Hindawi Publishing CorporationISRN Probability and StatisticsVolume 2013 Article ID 659580 14 pageshttpdxdoiorg1011552013659580

Research ArticleEstimates of Inequality Indices Based on Simple RandomRanked Set and Systematic Sampling

Pooja Bansal Sangeeta Arora and Kalpana K Mahajan

Department of Statistics Panjab University Chandigarh 160014 India

Correspondence should be addressed to Sangeeta Arora sarora131gmailcom

Received 16 June 2013 Accepted 2 August 2013

Academic Editors X Dang and S Lototsky

Copyright copy 2013 Pooja Bansal et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Gini index Bonferroni index and Absolute Lorenz index are some popular indices of inequality showing different features ofinequality measurement In general simple random sampling procedure is commonly used to estimate the inequality indices andtheir related inference The key condition that the samples must be drawn via simple random sampling procedure though makescalculations much simpler but this assumption is often violated in practice as the data does not always yield simple random sampleNonsimple random samples like Ranked set sampling or stratified sampling are gaining popularity for estimating these indicesThe purpose of the present paper is to compare the efficiency of simple random sample estimates of inequality indices with theirnonsimple random counterparts Monte Carlo simulation technique is applied to get the results for some specific distributions

1 Introduction

Lorenz curve [1] and the associated Gini index [2] are oneof the most popular and frequently used tools to measureincome inequality Certain other variants of Lorenz curvenamely Generalized Lorenz curve [3] Absolute Lorenz curve[4] Bonferroni curve [5] and Comic curves [3 6] andassociated inequality indices that is Generalized Lorenzindex Absolute Lorenz index Bonferroni index and Comicindex are some of the popular alternatives to Gini indexused to study certain specialized features of inequality Inpractice mostly the simple random sampling procedure isused to derive the statistical inference or obtaining the sampleestimates of these inequality measures The key conditionthat the samples must be drawn via simple random samplingprocedure though makes calculations much simpler but thisassumption is often violated in practice as the data does notalways yield simple random sample For example in Indiathe socioeconomic data collected by the National SampleSurvey Organization (NSSO) is not drawn through simplerandom sampling but follows two-stage stratified samplingtechniques Also in theUnited States commonly used incomeand earnings data such as the Current Population Survey

(CPS) and the Panel Study of Income Dynamics (PSID)are all multistage random samples where simple randomsampling is not the only method applied at each stage Onemay quote stratified sampling cluster sampling multistagecluster sampling and so forth as alternatives to simplerandom sampling while estimating inequality indices [7]

Besides these available traditional sampling methodsthe method of Ranked set sampling (RSS) have also gainedpopularity in the literature for estimating parameters likepopulation mean and variance and its associated statisticalinference This RSS method of sampling is proved to bemore efficient than simple random sampling [8] Thus theRSS technique may also be extended to obtain the sampleestimates of these inequalitymeasures and it will be extendedto compare the efficiency of RSS with simple random andother sampling estimates as applied to inequality measures[9] have given the comparison of ranked set sample estimateof Gini index with simple random sampling for a non-standard Lorenz curve equation using simulation techniqueand are not true in general as no standard distributions areconsidered

The purpose of the present work is to compare thenonsimple random estimates of the inequality measures

2 ISRN Probability and Statistics

Table 1 Simple random estimates of inequality indices

Inequality indices Simple random sampling estimates

Gini index (119866) 119866 = 1 minus2

119899

119899

sum

119894=1

119871 (119901119894)

Bonferroni index (119861) 119861 = 1 minus1

119899

119899

sum

119894=1

119861 (119901119894)

Absolute Lorenz index (119860) 119860 = minus1

119899

119899

sum

119894=1

AL (119901119894)

Table 2 Ranked set sample estimates of inequality indices

Inequality indices Ranked set sampling estimates

Gini index (119866) 119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871(119901(119894)119895)

119899

Bonferroni index (119861) 119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861(119901(119894)119895)

119899

Absolute Lorenz index (119860) 119860 = minus

sum119903

119895=1sum119898

119894=1AL(119901

(119894)119895)

119899

with that of simple random estimates In particular wehave applied two nonsimple random sampling techniquesmdashRanked set sampling and systematic sampling methods andhave obtained the sample estimates of some popular inequal-ity indicesmdashGini Bonferroni and Absolute Lorenz index viamethod of Monte Carlo simulation [10] and have comparedtheir efficiencies with those of simple random estimatesSimulated results showing the relative efficiencies have beenobtained for Pareto Exponential and power distribution Itis interesting to note that the simple random estimates havemuch less efficiency as compared to their nonsimple counter-parts that is ranked set or systematic sample estimates Wehave also compared the ranked set estimates with systematicsample estimates and found that systematic sample estimatesare more efficient than the ranked set sample estimates

The present paper has been organized in the followingmanner In Section 2 we have applied the estimates of somepopular inequality indicesmdashGini Index Bonferroni Indexand Absolute Lorenz Index for three sampling proceduresnamely simple random sampling ranked set sampling andsystematic samplingmdashand applied the method of MonteCarlo method of simulation to obtain the estimates A briefintroduction of the inequality indices and their estimationin case of three sampling techniques is given in this sectionSimulation study has been performed in Section 3 wherewe have computed the efficiencies of these estimates forsome specific distributions namely Exponential Pareto andPower distributions The table showing the various compar-ison of Mean Square Errors (MSE) under simple randomsampling ranked set sampling and systematic sampling isgiven at the end of Section 3 along with the conclusion

2 Monte Carlo Estimates for DifferentSampling Techniques

In this section we have discussed three sampling techniquesnamely simple random sampling ranked set sampling and

Table 3 Systematic sample estimates of inequality indices

Inequality indices Systematic random sampling estimates

Gini index (119866) 119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894)

Bonferroni index (119861) 119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894)

Absolute Lorenz index (119860) 119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894)

systematic sampling procedures and obtained the MonteCarlo estimates of the three indices Gini Bonferroni andAbsolute Lorenz

21 Simple Random Sampling Procedure The sample meanMonte Carlo procedure for the inequality indices usingsimple random sampling is explained below

211 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901 (1)

= 1 minus 2int

1

0

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901

= 1 minus 2119864[119871 (119901)

119891 (119901)]

= 1 minus 21198681

(2)

where

119871 (119901) =1

120583int

119901

0

119865minus1

(119905) 119889119905 0 le 119901 le 1 (3)

is the Lorenz curve at 119901 ordinates [1] and

119865minus1

(119901) = inf119909 119865 (119909) ge 119901

1198681= 119864[

119871 (119901)

119891 (119901)]

(4)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

119891 (119901) = 1 if 0 lt 119901 lt 10 ow

997904rArr 1198681= 119864 [119871 (119901)]

(5)

An unbiased estimator of 119868 is sample mean

1198681=

1

119899

119899

sum

119894=1

119871 (119901119894) (Reference [10]) (6)

where 119899 is the sample size

ISRN Probability and Statistics 3

Table 4 Gini index (119866) for given distributions

Distribution cdf Lorenz curve 119871(119901) Gini index (119866)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0 119901 + (1 minus 119901) log(1 minus 119901) 1

2

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0 1 minus (1 minus 119901)

(120572minus1)1205721

(2120572 minus 1)

Power 119909120572

120572 gt 1 0 le 119909 le 1 119901(120572+1)120572

1

(2120572 + 1)

Table 5 Bonferroni index (119861) for given distributions

Distribution cdf Bonferroni curve 119861(119901) Bonferroni index (119861)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0119901 + (1 minus 119901) log(1 minus 119901)

119901

1205872

6minus 1

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0

1 minus (1 minus 119901)(120572minus1)120572

119901120595 (2) minus 120595(

(120572 minus 1)

120572)

Power 119909120572

120572 gt 1 0 le 119909 le 1 1199011120572

1

(120572 + 1)

Where 120595(119909) is the digamma function

Therefore unbiased estimator of 119866 is

119866 = 1 minus 21198681 (7)

Using (6)

119866 = 1 minus2

119899

119899

sum

119894=1

119871 (119901119894) (8)

Var (119866) = var[1 minus 2

119899

119899

sum

119894=1

119871 (119901119894)]

=4

1198992var[119899

sum

119894=1

119871 (119901119894)]

=4

119899119864[119871 (119901)]

2

minus 1198682

1 from (5)

=4

119899[int

1

0

(119871 (119901))2

119889119901 minus (1 minus 119866

2)

2

] from (1)

(9)

Algorithm for computing simple random estimate for Giniindex is given as follows

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)

(ii) Compute 119901119894= 119883119894 119894 = 1 119873

(iii) Compute 119871(119901119894) 119894 = 1 119873

(iv) Compute 119866 = 1 minus (2119899)sum119899

119894=1119871(119901119894)

212 Bonferroni Curve Bonferroni index is defined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus int

1

0

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901

= 1 minus 119864[119861 (119901)

119891 (119901)]

= 1 minus 1198682

(10)

where

119861 (119901) =119871 (119901)

119901=

1

120583 sdot 119901int

119901

0

119865minus1

(119905) 119889119905 0 le 119901 le 1 (11)

is the Bonferroni curve defined at 119901 ordinates [5]

1198682= 119864[

119861 (119901)

119891 (119901)] (12)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198682= 119864 [119861 (119901)] (13)

An unbiased estimator of 119868 is sample mean

1198682=

1

119899

119899

sum

119894=1

119861 (119901119894) (14)

where 119899 is the sample sizeTherefore unbiased estimator of 119861 is

119861 = 1 minus 1198682 (15)

4 ISRN Probability and Statistics

Table 6 Absolute Lorenz index (119860) for given distributions

Distribution cdf Absolute Lorenz curve Absolute Lorenz index (119860)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0(1 minus 119901) log (1 minus 119901)

120582

1

4120582

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0

1205721199090(1 minus 119901) [1 minus (1 minus 119901)

minus1120572

]

(120572 minus 1)

1205721199090

2 (120572 minus 1) (2120572 minus 1)

Rectangular 119909 minus 119886

120579 119886 le 119909 le 119886 + 120579 120579 gt 0 119886 ge 0

120579119901(119901 minus 1)

2

120579

12

Table 7 Relative efficiency of Gini index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002498093 004457217 000910646 1784248 4894566 27432104 001530442 003360818 0004001166 2195979 8399597 38249905 001046073 002692506 0002174010 2573918 12384975 48117216 0007608975 002218654 0001295014 2915837 17132278 5875593

5

3 001002222 001747601 0003524347 1743726 4958652 28437104 000611229 001333912 0001584625 2182345 8417840 38572475 0004159828 001100573 00008643665 2645717 12732712 48125746 0003042587 000911744 00005193771 2996608 17554567 5858146

20

3 0002467123 000450777 00009057722 1827136 4976715 27237794 0001525546 0003313244 00004123536 2171842 8034958 36996065 0001024757 0002690909 00002184094 2625900 12320482 46919096 00007222134 0002298709 00001300434 3182866 17676476 5553634

100

3 00005033155 00008831595 00001777265 1754684 4969205 28319674 00003078470 00006709811 814811119890 minus 05 2179593 8234807 37781405 00002099437 00005397776 4331869119890 minus 05 2571059 12460617 48464926 00001524544 00004400118 2622736119890 minus 05 2886186 16776824 5812800

200

3 00002471209 00004459992 8781071119890 minus 05 1804781 5079098 28142464 0000153431 00003294056 4053465119890 minus 05 214693 8126519 37851815 00001031339 00002670593 2214463119890 minus 05 2589443 12059777 46572876 7600503119890 minus 05 00002178877 1308801119890 minus 05 2866754 16647886 5807226

500

3 00001002982 00001764347 3693063119890 minus 05 1759102 4777463 27158544 5943145119890 minus 05 00001320398 1595697119890 minus 05 2221716 8274741 37244825 4208938119890 minus 05 00001076813 88025119890 minus 06 2558395 12233036 47815266 3117683119890 minus 05 9065143119890 minus 05 5230148119890 minus 06 2907654 17332479 5960984

Using (14)

119861 = 1 minus1

119899

119899

sum

119894=1

119861 (119901119894) (16)

Var (119861) = var[1 minus 1

119899

119899

sum

119894=1

119861 (119901119894)]

=1

1198992var[119899

sum

119894=1

119861 (119901119894)]

=1

119899119864[119861 (119901)]

2

minus 1198682

2 from (13)

=1

119899[int

1

0

(119861 (119901))2

119889119901 minus (1 minus 119861)2

] from (10)

(17)

Algorithm for computing simple random estimate for Bon-ferroni index is given as follows

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)

(ii) Compute 119901119894= 119883119894 119894 = 1 119873

(iii) Compute 119861(119901119894) 119894 = 1 119873

(iv) Compute 119861 = 1 minus (1119899)sum119899119894=1

119861(119901119894)

ISRN Probability and Statistics 5

Table 8 Relative efficiency of Gini index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002332234 004529824 000589787 1942268 7680441 39543674 001381014 003470658 0002580012 2513122 13452100 53527435 0009193211 002779215 0001352259 3023116 20552387 67984106 000659181 002279101 00008011307 3457473 28448554 8228133

5

3 0009292055 001775528 0002399531 1910802 7399479 38724464 0005570443 001392200 0001050541 2499263 13252220 53024525 0003769394 001075287 00005584852 2852678 19253635 67493186 000268615 0009290222 00003206307 3458554 28974836 8377707

20

3 0002275199 0004430962 00006094044 1947505 7270971 37334804 0001390079 0003490160 00002649288 2510764 13173955 52469915 00009236206 0002648793 00001380824 2867836 19182698 66889096 00006634941 0002265676 8109345119890 minus 05 3414765 27939075 8181846

100

3 0000461317 000090545 00001197988 1962750 7558089 38507654 00002732137 00006699247 5275955119890 minus 05 2452017 12697695 51784695 00001887224 00005433437 2791159119890 minus 05 2879063 19466598 67614356 00001309059 00004580686 1628860119890 minus 05 3499220 28122036 8036658

200

3 00002253251 00004546124 6028802119890 minus 05 2017584 7540676 37374774 00001392043 00003422813 2601848119890 minus 05 2458842 13155315 53502095 9250598119890 minus 05 00002770614 1426628119890 minus 05 2995065 19420718 64842406 6674519119890 minus 05 00002295553 7895343119890 minus 06 3439278 29074772 8453742

500

3 9473933119890 minus 05 00001802806 2443929119890 minus 05 1902912 7376671 38765174 5606348119890 minus 05 00001383186 1074361119890 minus 05 2467178 12874499 52183095 3699474119890 minus 05 00001102943 5493003119890 minus 06 2981350 20079053 67348846 2701999119890 minus 05 9018714119890 minus 05 3301841119890 minus 06 3337793 27314198 8183310

213 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus int

1

0

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901

= minus 119864[AL (119901)119891 (119901)

]

= minus 1198683

(18)

where

AL (119901) = minus120583

2+ int

1

0

119909119865 (119909) 119889119865 (119909) (19)

is the Absolute Lorenz curve defined at 119901 ordinates [6]

1198683= 119864[

AL (119901)119891 (119901)

] (20)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198683= 119864 [AL (119901)] (21)

An unbiased estimator of 119868 is sample mean

1198683=1

119899

119899

sum

119894=1

AL (119901119894) (22)

where 119899 is the sample sizeTherefore unbiased estimator of 119860 is

119860 = minus1198683

(23)

using (22)

119860 = minus1

119899

119899

sum

119894=1

AL (119901119894) (24)

Var (119860) = var[minus1119899

119899

sum

119894=1

AL (119901119894)]

=1

1198992var[119899

sum

119894=1

AL (119901119894)]

=1

119899119864[AL (119901)]2 minus 1198682

3 from (21)

=1

119899[int

1

0

(AL (119901))2119889119901 minus 1198602] from (18)

(25)

Algorithm for computing simple random estimate for Abso-lute Lorenz index is given as follows

6 ISRN Probability and Statistics

Table 9 Relative efficiency of Gini index for power (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 003002883 005904002 0006465061 1966111 9132168 46447874 001837878 004518542 0002769992 2458565 16312473 66349585 001202134 003688425 0001390412 3068232 26527569 86458836 0008600471 002950916 0000838656 3431109 35186250 10255064

5

3 001183113 002385245 000263158 2016075 9063927 44958284 0007109205 001767874 0001112017 2486739 15897904 63930725 0004782672 001397249 00005790662 2921482 24129348 82592846 0003415356 001160659 00003343554 3398354 34713332 10214748

20

3 0003007737 0005933085 00006660064 1972607 8908450 45160784 0001856484 0004438954 00002774993 2391054 15996271 66900495 0001176903 0003597289 00001420534 3056573 25323498 82849346 00008649264 000293501 8396861119890 minus 05 3393364 34953657 10300592

100

3 00006040928 0001167984 00001285668 1933451 9084647 46986694 00003599259 00008980578 5426289119890 minus 05 2495118 16550128 66330035 00002358945 00007081798 2842227119890 minus 05 3002103 24916370 82996366 0000173568 00005906876 1640179119890 minus 05 3403206 36013606 10582260

200

3 00002970297 00005835807 6760029119890 minus 05 1964722 8632814 43939124 00001840207 00004393907 2766382119890 minus 05 2387724 15883226 66520355 00001199216 00003552514 1421686119890 minus 05 2962364 24988035 84351686 884773119890 minus 05 00002953183 821482119890 minus 06 3337786 35949455 10770449

500

3 00001217598 00002407522 2666487119890 minus 05 1977272 9028816 45663004 7261201119890 minus 05 00001784493 1124082119890 minus 05 2457572 15875114 64596725 48425119890 minus 05 00001436364 5589331119890 minus 06 2966163 25698317 86638286 3399731119890 minus 05 00001184066 3256773119890 minus 06 3482823 36357032 10438956

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)(ii) Compute 119901

119894= 119883119894 119894 = 1 119873

(iii) Compute AL(119901119894) 119894 = 1 119873

(iv) Compute 119860 = minus(1119899)sum119899

119894=1AL(119901119894)

Simple random sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in the Table 1

22 Ranked Set Sampling Procedure General ranked setsam-pling procedure involves the following steps [6]

(i) Select a simple random sample of 119898 units fromthe population and subject it to ordering on attribute ofinterestvia some ranking process that is

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

(26)

This judgment ranking can result from a variety ofmechanisms including expertrsquos opinion visual comparisonsor the use of easy-to-obtain auxiliary variables but it cannotinvolve actual measurements of the attribute of interest onsample units

Once this judgement ranking of 119898 units in our initialrandom sample has been accomplished the item judged to besmallest is included as the first item in the ranked set sampleand the attribute of interest is formally measured on this unit

and remaining (119898 minus 1) unmeasured units in the first randomsample are not considered further

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

997904rArr 119883[1]11

(27)

This measurement is denoted by 119883[1]11

the smallestorderstatistics or the smallest judgement order item It mayor may not actually have the smallest attribute measurementamong the 119898 sampled units Note that 119883

(2)11lt sdot sdot sdot lt 119883

(119898)11

are not considered further in the selection of our ranked setsampleThe sole purpose of (119898minus1) units is to help in selectingan item formeasurement that represents the smaller attributevalues in the population

(ii) Following selection of 119883[1]11

a second independentrandom sample of size 119898 is selected from the populationand judgement ranking without formal measurement on theattribute of interest is made and the second smallest item of119898 units in second random sample is selected and is includedin the ranked set sample for measurement of attribute ofinterest that is

119883(1)21

lt sdot sdot sdot lt 119883(119898)21

997904rArr 119883[2]21

(28)

The second measured observation is denoted by119883[2]21

(iii) Similarly from a third independent random sam-

ple we select the unit judgement ranked to be the thirdsmallest119883

[3]31 formeasurement and inclusion in the ranked-

set sample that is

119883(1)31

lt sdot sdot sdot lt 119883(119898)31

997904rArr 119883[3]31

(29)

ISRN Probability and Statistics 7

Table 10 Relative efficiency of Bonferroni index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 006202481 01235527 001377850 1991989 8967065 45015654 00381469 009533465 0005916627 2499146 16113007 64474075 002535770 007558111 0002995199 2980598 25234086 84661156 001791203 006181676 0001748279 3451131 35358636 10245521

5

3 002490822 005036193 0005549141 20219 9075626 44886624 001489808 003685433 0002358322 2473765 15627353 63172375 000982462 003053955 0001206127 3108471 25320344 81455936 0007220886 002487922 00006952421 3445452 35784973 10386146

20

3 0006252341 001245597 0001364699 1992209 9127265 45814804 0003685856 0009230949 00005877362 2504425 15705939 62712765 0002550649 0007501943 00002866664 2941190 26169593 88976216 0001808095 000619635 00001755058 3427005 35305671 10302195

100

3 0001249233 0002446855 00002789078 1958686 8772989 44790184 00007403018 0001914464 00001151496 2586059 16625885 64290445 0000504336 0001465721 601177119890 minus 05 2906238 24380856 83891436 00003529966 0001248915 3414360119890 minus 05 3538037 36578305 10338588

200

3 00006307183 0001248089 00001376957 1978838 9064110 45805234 00003722346 00009302265 5962156119890 minus 05 2499033 15602183 62432895 00002493340 00007560325 2945604119890 minus 05 3032207 25666468 84646146 00001767726 0000618749 1745216119890 minus 05 3500253 35454007 10128981

500

3 00002467436 00005050265 5529099119890 minus 05 2046767 9133975 44626374 00001511701 00003785248 2356102119890 minus 05 2503966 16065722 64161105 998555119890 minus 05 00002998347 1208355119890 minus 05 3002686 24813461 82637556 7085566119890 minus 05 0000252477 6991167119890 minus 06 3563258 36113713 10135026

This process is continued until we have selected the unitjudgement ranked to be largest of119898 units in the119898th randomsample denoted by 119883

[119898]1198981 that is 119883

(1)1198981lt sdot sdot sdot lt 119883

(119898)1198981rArr

119883[119898]1198981

for inclusion in our ranked set sampleThis entire process is referred to as a cycle and the number

of observations in each random sample119898 in our example iscalled set size

Thus to complete a single ranked set cycle we needto judge rank 119898 independent random samples of size 119898

involving a total of 1198982 sample units in order to obtain 119898

measured observations These 119898 observations represent abalanced ranked set sample with set size119898

In practice the sample size 119898 is kept small to ease thevisual ranking the RSS literature suggested 119898 = 2 3 4 5 or6 Therefore in order to obtain a ranked set sample with adesired total number of measured observation 119899 = 119903 lowast 119898 werepeat the entire cycle process 119903 independent times yieldingthe data

119883[1]119894119895

lt sdot sdot sdot lt 119883[119898]119894119895

119894 = 1 119898 119895 = 1 119903 (30)

Algorithm for obtaining the RSS estimate of Gini indexvia method of Monte Carlo simulation is given as follows

(a) Generate an RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119871(119901

(119894)119895)

(d) Compute the ranked set sample estimate ofGini indexas

119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871 (119901(119894)119895)

119899 (31)

Algorithm for obtaining the RSS estimate of Bonferroniindex via method of Monte Carlo simulation is given asfollowing

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119861(119901

(119894)119895)

(d) Compute the ranked set sample estimate of Bonfer-roni index as follows

119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861 (119901(119894)119895)

119899 (32)

Algorithm for obtaining the RSS estimate of AbsoluteLorenz index via method of Monte Carlo simulation is givenas follows

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Stochastic AnalysisInternational Journal of

2 ISRN Probability and Statistics

Table 1 Simple random estimates of inequality indices

Inequality indices Simple random sampling estimates

Gini index (119866) 119866 = 1 minus2

119899

119899

sum

119894=1

119871 (119901119894)

Bonferroni index (119861) 119861 = 1 minus1

119899

119899

sum

119894=1

119861 (119901119894)

Absolute Lorenz index (119860) 119860 = minus1

119899

119899

sum

119894=1

AL (119901119894)

Table 2 Ranked set sample estimates of inequality indices

Inequality indices Ranked set sampling estimates

Gini index (119866) 119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871(119901(119894)119895)

119899

Bonferroni index (119861) 119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861(119901(119894)119895)

119899

Absolute Lorenz index (119860) 119860 = minus

sum119903

119895=1sum119898

119894=1AL(119901

(119894)119895)

119899

with that of simple random estimates In particular wehave applied two nonsimple random sampling techniquesmdashRanked set sampling and systematic sampling methods andhave obtained the sample estimates of some popular inequal-ity indicesmdashGini Bonferroni and Absolute Lorenz index viamethod of Monte Carlo simulation [10] and have comparedtheir efficiencies with those of simple random estimatesSimulated results showing the relative efficiencies have beenobtained for Pareto Exponential and power distribution Itis interesting to note that the simple random estimates havemuch less efficiency as compared to their nonsimple counter-parts that is ranked set or systematic sample estimates Wehave also compared the ranked set estimates with systematicsample estimates and found that systematic sample estimatesare more efficient than the ranked set sample estimates

The present paper has been organized in the followingmanner In Section 2 we have applied the estimates of somepopular inequality indicesmdashGini Index Bonferroni Indexand Absolute Lorenz Index for three sampling proceduresnamely simple random sampling ranked set sampling andsystematic samplingmdashand applied the method of MonteCarlo method of simulation to obtain the estimates A briefintroduction of the inequality indices and their estimationin case of three sampling techniques is given in this sectionSimulation study has been performed in Section 3 wherewe have computed the efficiencies of these estimates forsome specific distributions namely Exponential Pareto andPower distributions The table showing the various compar-ison of Mean Square Errors (MSE) under simple randomsampling ranked set sampling and systematic sampling isgiven at the end of Section 3 along with the conclusion

2 Monte Carlo Estimates for DifferentSampling Techniques

In this section we have discussed three sampling techniquesnamely simple random sampling ranked set sampling and

Table 3 Systematic sample estimates of inequality indices

Inequality indices Systematic random sampling estimates

Gini index (119866) 119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894)

Bonferroni index (119861) 119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894)

Absolute Lorenz index (119860) 119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894)

systematic sampling procedures and obtained the MonteCarlo estimates of the three indices Gini Bonferroni andAbsolute Lorenz

21 Simple Random Sampling Procedure The sample meanMonte Carlo procedure for the inequality indices usingsimple random sampling is explained below

211 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901 (1)

= 1 minus 2int

1

0

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901

= 1 minus 2119864[119871 (119901)

119891 (119901)]

= 1 minus 21198681

(2)

where

119871 (119901) =1

120583int

119901

0

119865minus1

(119905) 119889119905 0 le 119901 le 1 (3)

is the Lorenz curve at 119901 ordinates [1] and

119865minus1

(119901) = inf119909 119865 (119909) ge 119901

1198681= 119864[

119871 (119901)

119891 (119901)]

(4)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

119891 (119901) = 1 if 0 lt 119901 lt 10 ow

997904rArr 1198681= 119864 [119871 (119901)]

(5)

An unbiased estimator of 119868 is sample mean

1198681=

1

119899

119899

sum

119894=1

119871 (119901119894) (Reference [10]) (6)

where 119899 is the sample size

ISRN Probability and Statistics 3

Table 4 Gini index (119866) for given distributions

Distribution cdf Lorenz curve 119871(119901) Gini index (119866)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0 119901 + (1 minus 119901) log(1 minus 119901) 1

2

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0 1 minus (1 minus 119901)

(120572minus1)1205721

(2120572 minus 1)

Power 119909120572

120572 gt 1 0 le 119909 le 1 119901(120572+1)120572

1

(2120572 + 1)

Table 5 Bonferroni index (119861) for given distributions

Distribution cdf Bonferroni curve 119861(119901) Bonferroni index (119861)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0119901 + (1 minus 119901) log(1 minus 119901)

119901

1205872

6minus 1

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0

1 minus (1 minus 119901)(120572minus1)120572

119901120595 (2) minus 120595(

(120572 minus 1)

120572)

Power 119909120572

120572 gt 1 0 le 119909 le 1 1199011120572

1

(120572 + 1)

Where 120595(119909) is the digamma function

Therefore unbiased estimator of 119866 is

119866 = 1 minus 21198681 (7)

Using (6)

119866 = 1 minus2

119899

119899

sum

119894=1

119871 (119901119894) (8)

Var (119866) = var[1 minus 2

119899

119899

sum

119894=1

119871 (119901119894)]

=4

1198992var[119899

sum

119894=1

119871 (119901119894)]

=4

119899119864[119871 (119901)]

2

minus 1198682

1 from (5)

=4

119899[int

1

0

(119871 (119901))2

119889119901 minus (1 minus 119866

2)

2

] from (1)

(9)

Algorithm for computing simple random estimate for Giniindex is given as follows

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)

(ii) Compute 119901119894= 119883119894 119894 = 1 119873

(iii) Compute 119871(119901119894) 119894 = 1 119873

(iv) Compute 119866 = 1 minus (2119899)sum119899

119894=1119871(119901119894)

212 Bonferroni Curve Bonferroni index is defined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus int

1

0

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901

= 1 minus 119864[119861 (119901)

119891 (119901)]

= 1 minus 1198682

(10)

where

119861 (119901) =119871 (119901)

119901=

1

120583 sdot 119901int

119901

0

119865minus1

(119905) 119889119905 0 le 119901 le 1 (11)

is the Bonferroni curve defined at 119901 ordinates [5]

1198682= 119864[

119861 (119901)

119891 (119901)] (12)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198682= 119864 [119861 (119901)] (13)

An unbiased estimator of 119868 is sample mean

1198682=

1

119899

119899

sum

119894=1

119861 (119901119894) (14)

where 119899 is the sample sizeTherefore unbiased estimator of 119861 is

119861 = 1 minus 1198682 (15)

4 ISRN Probability and Statistics

Table 6 Absolute Lorenz index (119860) for given distributions

Distribution cdf Absolute Lorenz curve Absolute Lorenz index (119860)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0(1 minus 119901) log (1 minus 119901)

120582

1

4120582

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0

1205721199090(1 minus 119901) [1 minus (1 minus 119901)

minus1120572

]

(120572 minus 1)

1205721199090

2 (120572 minus 1) (2120572 minus 1)

Rectangular 119909 minus 119886

120579 119886 le 119909 le 119886 + 120579 120579 gt 0 119886 ge 0

120579119901(119901 minus 1)

2

120579

12

Table 7 Relative efficiency of Gini index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002498093 004457217 000910646 1784248 4894566 27432104 001530442 003360818 0004001166 2195979 8399597 38249905 001046073 002692506 0002174010 2573918 12384975 48117216 0007608975 002218654 0001295014 2915837 17132278 5875593

5

3 001002222 001747601 0003524347 1743726 4958652 28437104 000611229 001333912 0001584625 2182345 8417840 38572475 0004159828 001100573 00008643665 2645717 12732712 48125746 0003042587 000911744 00005193771 2996608 17554567 5858146

20

3 0002467123 000450777 00009057722 1827136 4976715 27237794 0001525546 0003313244 00004123536 2171842 8034958 36996065 0001024757 0002690909 00002184094 2625900 12320482 46919096 00007222134 0002298709 00001300434 3182866 17676476 5553634

100

3 00005033155 00008831595 00001777265 1754684 4969205 28319674 00003078470 00006709811 814811119890 minus 05 2179593 8234807 37781405 00002099437 00005397776 4331869119890 minus 05 2571059 12460617 48464926 00001524544 00004400118 2622736119890 minus 05 2886186 16776824 5812800

200

3 00002471209 00004459992 8781071119890 minus 05 1804781 5079098 28142464 0000153431 00003294056 4053465119890 minus 05 214693 8126519 37851815 00001031339 00002670593 2214463119890 minus 05 2589443 12059777 46572876 7600503119890 minus 05 00002178877 1308801119890 minus 05 2866754 16647886 5807226

500

3 00001002982 00001764347 3693063119890 minus 05 1759102 4777463 27158544 5943145119890 minus 05 00001320398 1595697119890 minus 05 2221716 8274741 37244825 4208938119890 minus 05 00001076813 88025119890 minus 06 2558395 12233036 47815266 3117683119890 minus 05 9065143119890 minus 05 5230148119890 minus 06 2907654 17332479 5960984

Using (14)

119861 = 1 minus1

119899

119899

sum

119894=1

119861 (119901119894) (16)

Var (119861) = var[1 minus 1

119899

119899

sum

119894=1

119861 (119901119894)]

=1

1198992var[119899

sum

119894=1

119861 (119901119894)]

=1

119899119864[119861 (119901)]

2

minus 1198682

2 from (13)

=1

119899[int

1

0

(119861 (119901))2

119889119901 minus (1 minus 119861)2

] from (10)

(17)

Algorithm for computing simple random estimate for Bon-ferroni index is given as follows

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)

(ii) Compute 119901119894= 119883119894 119894 = 1 119873

(iii) Compute 119861(119901119894) 119894 = 1 119873

(iv) Compute 119861 = 1 minus (1119899)sum119899119894=1

119861(119901119894)

ISRN Probability and Statistics 5

Table 8 Relative efficiency of Gini index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002332234 004529824 000589787 1942268 7680441 39543674 001381014 003470658 0002580012 2513122 13452100 53527435 0009193211 002779215 0001352259 3023116 20552387 67984106 000659181 002279101 00008011307 3457473 28448554 8228133

5

3 0009292055 001775528 0002399531 1910802 7399479 38724464 0005570443 001392200 0001050541 2499263 13252220 53024525 0003769394 001075287 00005584852 2852678 19253635 67493186 000268615 0009290222 00003206307 3458554 28974836 8377707

20

3 0002275199 0004430962 00006094044 1947505 7270971 37334804 0001390079 0003490160 00002649288 2510764 13173955 52469915 00009236206 0002648793 00001380824 2867836 19182698 66889096 00006634941 0002265676 8109345119890 minus 05 3414765 27939075 8181846

100

3 0000461317 000090545 00001197988 1962750 7558089 38507654 00002732137 00006699247 5275955119890 minus 05 2452017 12697695 51784695 00001887224 00005433437 2791159119890 minus 05 2879063 19466598 67614356 00001309059 00004580686 1628860119890 minus 05 3499220 28122036 8036658

200

3 00002253251 00004546124 6028802119890 minus 05 2017584 7540676 37374774 00001392043 00003422813 2601848119890 minus 05 2458842 13155315 53502095 9250598119890 minus 05 00002770614 1426628119890 minus 05 2995065 19420718 64842406 6674519119890 minus 05 00002295553 7895343119890 minus 06 3439278 29074772 8453742

500

3 9473933119890 minus 05 00001802806 2443929119890 minus 05 1902912 7376671 38765174 5606348119890 minus 05 00001383186 1074361119890 minus 05 2467178 12874499 52183095 3699474119890 minus 05 00001102943 5493003119890 minus 06 2981350 20079053 67348846 2701999119890 minus 05 9018714119890 minus 05 3301841119890 minus 06 3337793 27314198 8183310

213 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus int

1

0

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901

= minus 119864[AL (119901)119891 (119901)

]

= minus 1198683

(18)

where

AL (119901) = minus120583

2+ int

1

0

119909119865 (119909) 119889119865 (119909) (19)

is the Absolute Lorenz curve defined at 119901 ordinates [6]

1198683= 119864[

AL (119901)119891 (119901)

] (20)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198683= 119864 [AL (119901)] (21)

An unbiased estimator of 119868 is sample mean

1198683=1

119899

119899

sum

119894=1

AL (119901119894) (22)

where 119899 is the sample sizeTherefore unbiased estimator of 119860 is

119860 = minus1198683

(23)

using (22)

119860 = minus1

119899

119899

sum

119894=1

AL (119901119894) (24)

Var (119860) = var[minus1119899

119899

sum

119894=1

AL (119901119894)]

=1

1198992var[119899

sum

119894=1

AL (119901119894)]

=1

119899119864[AL (119901)]2 minus 1198682

3 from (21)

=1

119899[int

1

0

(AL (119901))2119889119901 minus 1198602] from (18)

(25)

Algorithm for computing simple random estimate for Abso-lute Lorenz index is given as follows

6 ISRN Probability and Statistics

Table 9 Relative efficiency of Gini index for power (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 003002883 005904002 0006465061 1966111 9132168 46447874 001837878 004518542 0002769992 2458565 16312473 66349585 001202134 003688425 0001390412 3068232 26527569 86458836 0008600471 002950916 0000838656 3431109 35186250 10255064

5

3 001183113 002385245 000263158 2016075 9063927 44958284 0007109205 001767874 0001112017 2486739 15897904 63930725 0004782672 001397249 00005790662 2921482 24129348 82592846 0003415356 001160659 00003343554 3398354 34713332 10214748

20

3 0003007737 0005933085 00006660064 1972607 8908450 45160784 0001856484 0004438954 00002774993 2391054 15996271 66900495 0001176903 0003597289 00001420534 3056573 25323498 82849346 00008649264 000293501 8396861119890 minus 05 3393364 34953657 10300592

100

3 00006040928 0001167984 00001285668 1933451 9084647 46986694 00003599259 00008980578 5426289119890 minus 05 2495118 16550128 66330035 00002358945 00007081798 2842227119890 minus 05 3002103 24916370 82996366 0000173568 00005906876 1640179119890 minus 05 3403206 36013606 10582260

200

3 00002970297 00005835807 6760029119890 minus 05 1964722 8632814 43939124 00001840207 00004393907 2766382119890 minus 05 2387724 15883226 66520355 00001199216 00003552514 1421686119890 minus 05 2962364 24988035 84351686 884773119890 minus 05 00002953183 821482119890 minus 06 3337786 35949455 10770449

500

3 00001217598 00002407522 2666487119890 minus 05 1977272 9028816 45663004 7261201119890 minus 05 00001784493 1124082119890 minus 05 2457572 15875114 64596725 48425119890 minus 05 00001436364 5589331119890 minus 06 2966163 25698317 86638286 3399731119890 minus 05 00001184066 3256773119890 minus 06 3482823 36357032 10438956

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)(ii) Compute 119901

119894= 119883119894 119894 = 1 119873

(iii) Compute AL(119901119894) 119894 = 1 119873

(iv) Compute 119860 = minus(1119899)sum119899

119894=1AL(119901119894)

Simple random sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in the Table 1

22 Ranked Set Sampling Procedure General ranked setsam-pling procedure involves the following steps [6]

(i) Select a simple random sample of 119898 units fromthe population and subject it to ordering on attribute ofinterestvia some ranking process that is

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

(26)

This judgment ranking can result from a variety ofmechanisms including expertrsquos opinion visual comparisonsor the use of easy-to-obtain auxiliary variables but it cannotinvolve actual measurements of the attribute of interest onsample units

Once this judgement ranking of 119898 units in our initialrandom sample has been accomplished the item judged to besmallest is included as the first item in the ranked set sampleand the attribute of interest is formally measured on this unit

and remaining (119898 minus 1) unmeasured units in the first randomsample are not considered further

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

997904rArr 119883[1]11

(27)

This measurement is denoted by 119883[1]11

the smallestorderstatistics or the smallest judgement order item It mayor may not actually have the smallest attribute measurementamong the 119898 sampled units Note that 119883

(2)11lt sdot sdot sdot lt 119883

(119898)11

are not considered further in the selection of our ranked setsampleThe sole purpose of (119898minus1) units is to help in selectingan item formeasurement that represents the smaller attributevalues in the population

(ii) Following selection of 119883[1]11

a second independentrandom sample of size 119898 is selected from the populationand judgement ranking without formal measurement on theattribute of interest is made and the second smallest item of119898 units in second random sample is selected and is includedin the ranked set sample for measurement of attribute ofinterest that is

119883(1)21

lt sdot sdot sdot lt 119883(119898)21

997904rArr 119883[2]21

(28)

The second measured observation is denoted by119883[2]21

(iii) Similarly from a third independent random sam-

ple we select the unit judgement ranked to be the thirdsmallest119883

[3]31 formeasurement and inclusion in the ranked-

set sample that is

119883(1)31

lt sdot sdot sdot lt 119883(119898)31

997904rArr 119883[3]31

(29)

ISRN Probability and Statistics 7

Table 10 Relative efficiency of Bonferroni index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 006202481 01235527 001377850 1991989 8967065 45015654 00381469 009533465 0005916627 2499146 16113007 64474075 002535770 007558111 0002995199 2980598 25234086 84661156 001791203 006181676 0001748279 3451131 35358636 10245521

5

3 002490822 005036193 0005549141 20219 9075626 44886624 001489808 003685433 0002358322 2473765 15627353 63172375 000982462 003053955 0001206127 3108471 25320344 81455936 0007220886 002487922 00006952421 3445452 35784973 10386146

20

3 0006252341 001245597 0001364699 1992209 9127265 45814804 0003685856 0009230949 00005877362 2504425 15705939 62712765 0002550649 0007501943 00002866664 2941190 26169593 88976216 0001808095 000619635 00001755058 3427005 35305671 10302195

100

3 0001249233 0002446855 00002789078 1958686 8772989 44790184 00007403018 0001914464 00001151496 2586059 16625885 64290445 0000504336 0001465721 601177119890 minus 05 2906238 24380856 83891436 00003529966 0001248915 3414360119890 minus 05 3538037 36578305 10338588

200

3 00006307183 0001248089 00001376957 1978838 9064110 45805234 00003722346 00009302265 5962156119890 minus 05 2499033 15602183 62432895 00002493340 00007560325 2945604119890 minus 05 3032207 25666468 84646146 00001767726 0000618749 1745216119890 minus 05 3500253 35454007 10128981

500

3 00002467436 00005050265 5529099119890 minus 05 2046767 9133975 44626374 00001511701 00003785248 2356102119890 minus 05 2503966 16065722 64161105 998555119890 minus 05 00002998347 1208355119890 minus 05 3002686 24813461 82637556 7085566119890 minus 05 0000252477 6991167119890 minus 06 3563258 36113713 10135026

This process is continued until we have selected the unitjudgement ranked to be largest of119898 units in the119898th randomsample denoted by 119883

[119898]1198981 that is 119883

(1)1198981lt sdot sdot sdot lt 119883

(119898)1198981rArr

119883[119898]1198981

for inclusion in our ranked set sampleThis entire process is referred to as a cycle and the number

of observations in each random sample119898 in our example iscalled set size

Thus to complete a single ranked set cycle we needto judge rank 119898 independent random samples of size 119898

involving a total of 1198982 sample units in order to obtain 119898

measured observations These 119898 observations represent abalanced ranked set sample with set size119898

In practice the sample size 119898 is kept small to ease thevisual ranking the RSS literature suggested 119898 = 2 3 4 5 or6 Therefore in order to obtain a ranked set sample with adesired total number of measured observation 119899 = 119903 lowast 119898 werepeat the entire cycle process 119903 independent times yieldingthe data

119883[1]119894119895

lt sdot sdot sdot lt 119883[119898]119894119895

119894 = 1 119898 119895 = 1 119903 (30)

Algorithm for obtaining the RSS estimate of Gini indexvia method of Monte Carlo simulation is given as follows

(a) Generate an RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119871(119901

(119894)119895)

(d) Compute the ranked set sample estimate ofGini indexas

119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871 (119901(119894)119895)

119899 (31)

Algorithm for obtaining the RSS estimate of Bonferroniindex via method of Monte Carlo simulation is given asfollowing

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119861(119901

(119894)119895)

(d) Compute the ranked set sample estimate of Bonfer-roni index as follows

119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861 (119901(119894)119895)

119899 (32)

Algorithm for obtaining the RSS estimate of AbsoluteLorenz index via method of Monte Carlo simulation is givenas follows

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 3

Table 4 Gini index (119866) for given distributions

Distribution cdf Lorenz curve 119871(119901) Gini index (119866)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0 119901 + (1 minus 119901) log(1 minus 119901) 1

2

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0 1 minus (1 minus 119901)

(120572minus1)1205721

(2120572 minus 1)

Power 119909120572

120572 gt 1 0 le 119909 le 1 119901(120572+1)120572

1

(2120572 + 1)

Table 5 Bonferroni index (119861) for given distributions

Distribution cdf Bonferroni curve 119861(119901) Bonferroni index (119861)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0119901 + (1 minus 119901) log(1 minus 119901)

119901

1205872

6minus 1

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0

1 minus (1 minus 119901)(120572minus1)120572

119901120595 (2) minus 120595(

(120572 minus 1)

120572)

Power 119909120572

120572 gt 1 0 le 119909 le 1 1199011120572

1

(120572 + 1)

Where 120595(119909) is the digamma function

Therefore unbiased estimator of 119866 is

119866 = 1 minus 21198681 (7)

Using (6)

119866 = 1 minus2

119899

119899

sum

119894=1

119871 (119901119894) (8)

Var (119866) = var[1 minus 2

119899

119899

sum

119894=1

119871 (119901119894)]

=4

1198992var[119899

sum

119894=1

119871 (119901119894)]

=4

119899119864[119871 (119901)]

2

minus 1198682

1 from (5)

=4

119899[int

1

0

(119871 (119901))2

119889119901 minus (1 minus 119866

2)

2

] from (1)

(9)

Algorithm for computing simple random estimate for Giniindex is given as follows

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)

(ii) Compute 119901119894= 119883119894 119894 = 1 119873

(iii) Compute 119871(119901119894) 119894 = 1 119873

(iv) Compute 119866 = 1 minus (2119899)sum119899

119894=1119871(119901119894)

212 Bonferroni Curve Bonferroni index is defined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus int

1

0

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901

= 1 minus 119864[119861 (119901)

119891 (119901)]

= 1 minus 1198682

(10)

where

119861 (119901) =119871 (119901)

119901=

1

120583 sdot 119901int

119901

0

119865minus1

(119905) 119889119905 0 le 119901 le 1 (11)

is the Bonferroni curve defined at 119901 ordinates [5]

1198682= 119864[

119861 (119901)

119891 (119901)] (12)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198682= 119864 [119861 (119901)] (13)

An unbiased estimator of 119868 is sample mean

1198682=

1

119899

119899

sum

119894=1

119861 (119901119894) (14)

where 119899 is the sample sizeTherefore unbiased estimator of 119861 is

119861 = 1 minus 1198682 (15)

4 ISRN Probability and Statistics

Table 6 Absolute Lorenz index (119860) for given distributions

Distribution cdf Absolute Lorenz curve Absolute Lorenz index (119860)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0(1 minus 119901) log (1 minus 119901)

120582

1

4120582

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0

1205721199090(1 minus 119901) [1 minus (1 minus 119901)

minus1120572

]

(120572 minus 1)

1205721199090

2 (120572 minus 1) (2120572 minus 1)

Rectangular 119909 minus 119886

120579 119886 le 119909 le 119886 + 120579 120579 gt 0 119886 ge 0

120579119901(119901 minus 1)

2

120579

12

Table 7 Relative efficiency of Gini index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002498093 004457217 000910646 1784248 4894566 27432104 001530442 003360818 0004001166 2195979 8399597 38249905 001046073 002692506 0002174010 2573918 12384975 48117216 0007608975 002218654 0001295014 2915837 17132278 5875593

5

3 001002222 001747601 0003524347 1743726 4958652 28437104 000611229 001333912 0001584625 2182345 8417840 38572475 0004159828 001100573 00008643665 2645717 12732712 48125746 0003042587 000911744 00005193771 2996608 17554567 5858146

20

3 0002467123 000450777 00009057722 1827136 4976715 27237794 0001525546 0003313244 00004123536 2171842 8034958 36996065 0001024757 0002690909 00002184094 2625900 12320482 46919096 00007222134 0002298709 00001300434 3182866 17676476 5553634

100

3 00005033155 00008831595 00001777265 1754684 4969205 28319674 00003078470 00006709811 814811119890 minus 05 2179593 8234807 37781405 00002099437 00005397776 4331869119890 minus 05 2571059 12460617 48464926 00001524544 00004400118 2622736119890 minus 05 2886186 16776824 5812800

200

3 00002471209 00004459992 8781071119890 minus 05 1804781 5079098 28142464 0000153431 00003294056 4053465119890 minus 05 214693 8126519 37851815 00001031339 00002670593 2214463119890 minus 05 2589443 12059777 46572876 7600503119890 minus 05 00002178877 1308801119890 minus 05 2866754 16647886 5807226

500

3 00001002982 00001764347 3693063119890 minus 05 1759102 4777463 27158544 5943145119890 minus 05 00001320398 1595697119890 minus 05 2221716 8274741 37244825 4208938119890 minus 05 00001076813 88025119890 minus 06 2558395 12233036 47815266 3117683119890 minus 05 9065143119890 minus 05 5230148119890 minus 06 2907654 17332479 5960984

Using (14)

119861 = 1 minus1

119899

119899

sum

119894=1

119861 (119901119894) (16)

Var (119861) = var[1 minus 1

119899

119899

sum

119894=1

119861 (119901119894)]

=1

1198992var[119899

sum

119894=1

119861 (119901119894)]

=1

119899119864[119861 (119901)]

2

minus 1198682

2 from (13)

=1

119899[int

1

0

(119861 (119901))2

119889119901 minus (1 minus 119861)2

] from (10)

(17)

Algorithm for computing simple random estimate for Bon-ferroni index is given as follows

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)

(ii) Compute 119901119894= 119883119894 119894 = 1 119873

(iii) Compute 119861(119901119894) 119894 = 1 119873

(iv) Compute 119861 = 1 minus (1119899)sum119899119894=1

119861(119901119894)

ISRN Probability and Statistics 5

Table 8 Relative efficiency of Gini index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002332234 004529824 000589787 1942268 7680441 39543674 001381014 003470658 0002580012 2513122 13452100 53527435 0009193211 002779215 0001352259 3023116 20552387 67984106 000659181 002279101 00008011307 3457473 28448554 8228133

5

3 0009292055 001775528 0002399531 1910802 7399479 38724464 0005570443 001392200 0001050541 2499263 13252220 53024525 0003769394 001075287 00005584852 2852678 19253635 67493186 000268615 0009290222 00003206307 3458554 28974836 8377707

20

3 0002275199 0004430962 00006094044 1947505 7270971 37334804 0001390079 0003490160 00002649288 2510764 13173955 52469915 00009236206 0002648793 00001380824 2867836 19182698 66889096 00006634941 0002265676 8109345119890 minus 05 3414765 27939075 8181846

100

3 0000461317 000090545 00001197988 1962750 7558089 38507654 00002732137 00006699247 5275955119890 minus 05 2452017 12697695 51784695 00001887224 00005433437 2791159119890 minus 05 2879063 19466598 67614356 00001309059 00004580686 1628860119890 minus 05 3499220 28122036 8036658

200

3 00002253251 00004546124 6028802119890 minus 05 2017584 7540676 37374774 00001392043 00003422813 2601848119890 minus 05 2458842 13155315 53502095 9250598119890 minus 05 00002770614 1426628119890 minus 05 2995065 19420718 64842406 6674519119890 minus 05 00002295553 7895343119890 minus 06 3439278 29074772 8453742

500

3 9473933119890 minus 05 00001802806 2443929119890 minus 05 1902912 7376671 38765174 5606348119890 minus 05 00001383186 1074361119890 minus 05 2467178 12874499 52183095 3699474119890 minus 05 00001102943 5493003119890 minus 06 2981350 20079053 67348846 2701999119890 minus 05 9018714119890 minus 05 3301841119890 minus 06 3337793 27314198 8183310

213 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus int

1

0

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901

= minus 119864[AL (119901)119891 (119901)

]

= minus 1198683

(18)

where

AL (119901) = minus120583

2+ int

1

0

119909119865 (119909) 119889119865 (119909) (19)

is the Absolute Lorenz curve defined at 119901 ordinates [6]

1198683= 119864[

AL (119901)119891 (119901)

] (20)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198683= 119864 [AL (119901)] (21)

An unbiased estimator of 119868 is sample mean

1198683=1

119899

119899

sum

119894=1

AL (119901119894) (22)

where 119899 is the sample sizeTherefore unbiased estimator of 119860 is

119860 = minus1198683

(23)

using (22)

119860 = minus1

119899

119899

sum

119894=1

AL (119901119894) (24)

Var (119860) = var[minus1119899

119899

sum

119894=1

AL (119901119894)]

=1

1198992var[119899

sum

119894=1

AL (119901119894)]

=1

119899119864[AL (119901)]2 minus 1198682

3 from (21)

=1

119899[int

1

0

(AL (119901))2119889119901 minus 1198602] from (18)

(25)

Algorithm for computing simple random estimate for Abso-lute Lorenz index is given as follows

6 ISRN Probability and Statistics

Table 9 Relative efficiency of Gini index for power (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 003002883 005904002 0006465061 1966111 9132168 46447874 001837878 004518542 0002769992 2458565 16312473 66349585 001202134 003688425 0001390412 3068232 26527569 86458836 0008600471 002950916 0000838656 3431109 35186250 10255064

5

3 001183113 002385245 000263158 2016075 9063927 44958284 0007109205 001767874 0001112017 2486739 15897904 63930725 0004782672 001397249 00005790662 2921482 24129348 82592846 0003415356 001160659 00003343554 3398354 34713332 10214748

20

3 0003007737 0005933085 00006660064 1972607 8908450 45160784 0001856484 0004438954 00002774993 2391054 15996271 66900495 0001176903 0003597289 00001420534 3056573 25323498 82849346 00008649264 000293501 8396861119890 minus 05 3393364 34953657 10300592

100

3 00006040928 0001167984 00001285668 1933451 9084647 46986694 00003599259 00008980578 5426289119890 minus 05 2495118 16550128 66330035 00002358945 00007081798 2842227119890 minus 05 3002103 24916370 82996366 0000173568 00005906876 1640179119890 minus 05 3403206 36013606 10582260

200

3 00002970297 00005835807 6760029119890 minus 05 1964722 8632814 43939124 00001840207 00004393907 2766382119890 minus 05 2387724 15883226 66520355 00001199216 00003552514 1421686119890 minus 05 2962364 24988035 84351686 884773119890 minus 05 00002953183 821482119890 minus 06 3337786 35949455 10770449

500

3 00001217598 00002407522 2666487119890 minus 05 1977272 9028816 45663004 7261201119890 minus 05 00001784493 1124082119890 minus 05 2457572 15875114 64596725 48425119890 minus 05 00001436364 5589331119890 minus 06 2966163 25698317 86638286 3399731119890 minus 05 00001184066 3256773119890 minus 06 3482823 36357032 10438956

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)(ii) Compute 119901

119894= 119883119894 119894 = 1 119873

(iii) Compute AL(119901119894) 119894 = 1 119873

(iv) Compute 119860 = minus(1119899)sum119899

119894=1AL(119901119894)

Simple random sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in the Table 1

22 Ranked Set Sampling Procedure General ranked setsam-pling procedure involves the following steps [6]

(i) Select a simple random sample of 119898 units fromthe population and subject it to ordering on attribute ofinterestvia some ranking process that is

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

(26)

This judgment ranking can result from a variety ofmechanisms including expertrsquos opinion visual comparisonsor the use of easy-to-obtain auxiliary variables but it cannotinvolve actual measurements of the attribute of interest onsample units

Once this judgement ranking of 119898 units in our initialrandom sample has been accomplished the item judged to besmallest is included as the first item in the ranked set sampleand the attribute of interest is formally measured on this unit

and remaining (119898 minus 1) unmeasured units in the first randomsample are not considered further

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

997904rArr 119883[1]11

(27)

This measurement is denoted by 119883[1]11

the smallestorderstatistics or the smallest judgement order item It mayor may not actually have the smallest attribute measurementamong the 119898 sampled units Note that 119883

(2)11lt sdot sdot sdot lt 119883

(119898)11

are not considered further in the selection of our ranked setsampleThe sole purpose of (119898minus1) units is to help in selectingan item formeasurement that represents the smaller attributevalues in the population

(ii) Following selection of 119883[1]11

a second independentrandom sample of size 119898 is selected from the populationand judgement ranking without formal measurement on theattribute of interest is made and the second smallest item of119898 units in second random sample is selected and is includedin the ranked set sample for measurement of attribute ofinterest that is

119883(1)21

lt sdot sdot sdot lt 119883(119898)21

997904rArr 119883[2]21

(28)

The second measured observation is denoted by119883[2]21

(iii) Similarly from a third independent random sam-

ple we select the unit judgement ranked to be the thirdsmallest119883

[3]31 formeasurement and inclusion in the ranked-

set sample that is

119883(1)31

lt sdot sdot sdot lt 119883(119898)31

997904rArr 119883[3]31

(29)

ISRN Probability and Statistics 7

Table 10 Relative efficiency of Bonferroni index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 006202481 01235527 001377850 1991989 8967065 45015654 00381469 009533465 0005916627 2499146 16113007 64474075 002535770 007558111 0002995199 2980598 25234086 84661156 001791203 006181676 0001748279 3451131 35358636 10245521

5

3 002490822 005036193 0005549141 20219 9075626 44886624 001489808 003685433 0002358322 2473765 15627353 63172375 000982462 003053955 0001206127 3108471 25320344 81455936 0007220886 002487922 00006952421 3445452 35784973 10386146

20

3 0006252341 001245597 0001364699 1992209 9127265 45814804 0003685856 0009230949 00005877362 2504425 15705939 62712765 0002550649 0007501943 00002866664 2941190 26169593 88976216 0001808095 000619635 00001755058 3427005 35305671 10302195

100

3 0001249233 0002446855 00002789078 1958686 8772989 44790184 00007403018 0001914464 00001151496 2586059 16625885 64290445 0000504336 0001465721 601177119890 minus 05 2906238 24380856 83891436 00003529966 0001248915 3414360119890 minus 05 3538037 36578305 10338588

200

3 00006307183 0001248089 00001376957 1978838 9064110 45805234 00003722346 00009302265 5962156119890 minus 05 2499033 15602183 62432895 00002493340 00007560325 2945604119890 minus 05 3032207 25666468 84646146 00001767726 0000618749 1745216119890 minus 05 3500253 35454007 10128981

500

3 00002467436 00005050265 5529099119890 minus 05 2046767 9133975 44626374 00001511701 00003785248 2356102119890 minus 05 2503966 16065722 64161105 998555119890 minus 05 00002998347 1208355119890 minus 05 3002686 24813461 82637556 7085566119890 minus 05 0000252477 6991167119890 minus 06 3563258 36113713 10135026

This process is continued until we have selected the unitjudgement ranked to be largest of119898 units in the119898th randomsample denoted by 119883

[119898]1198981 that is 119883

(1)1198981lt sdot sdot sdot lt 119883

(119898)1198981rArr

119883[119898]1198981

for inclusion in our ranked set sampleThis entire process is referred to as a cycle and the number

of observations in each random sample119898 in our example iscalled set size

Thus to complete a single ranked set cycle we needto judge rank 119898 independent random samples of size 119898

involving a total of 1198982 sample units in order to obtain 119898

measured observations These 119898 observations represent abalanced ranked set sample with set size119898

In practice the sample size 119898 is kept small to ease thevisual ranking the RSS literature suggested 119898 = 2 3 4 5 or6 Therefore in order to obtain a ranked set sample with adesired total number of measured observation 119899 = 119903 lowast 119898 werepeat the entire cycle process 119903 independent times yieldingthe data

119883[1]119894119895

lt sdot sdot sdot lt 119883[119898]119894119895

119894 = 1 119898 119895 = 1 119903 (30)

Algorithm for obtaining the RSS estimate of Gini indexvia method of Monte Carlo simulation is given as follows

(a) Generate an RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119871(119901

(119894)119895)

(d) Compute the ranked set sample estimate ofGini indexas

119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871 (119901(119894)119895)

119899 (31)

Algorithm for obtaining the RSS estimate of Bonferroniindex via method of Monte Carlo simulation is given asfollowing

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119861(119901

(119894)119895)

(d) Compute the ranked set sample estimate of Bonfer-roni index as follows

119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861 (119901(119894)119895)

119899 (32)

Algorithm for obtaining the RSS estimate of AbsoluteLorenz index via method of Monte Carlo simulation is givenas follows

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Stochastic AnalysisInternational Journal of

4 ISRN Probability and Statistics

Table 6 Absolute Lorenz index (119860) for given distributions

Distribution cdf Absolute Lorenz curve Absolute Lorenz index (119860)

Exponential 1 minus 119890minus120582119909

120582 gt 0 119909 ge 0(1 minus 119901) log (1 minus 119901)

120582

1

4120582

Pareto 1 minus (119909

1199090

)

minus120572

120572 gt 1 119909 ge 1199090gt 0

1205721199090(1 minus 119901) [1 minus (1 minus 119901)

minus1120572

]

(120572 minus 1)

1205721199090

2 (120572 minus 1) (2120572 minus 1)

Rectangular 119909 minus 119886

120579 119886 le 119909 le 119886 + 120579 120579 gt 0 119886 ge 0

120579119901(119901 minus 1)

2

120579

12

Table 7 Relative efficiency of Gini index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002498093 004457217 000910646 1784248 4894566 27432104 001530442 003360818 0004001166 2195979 8399597 38249905 001046073 002692506 0002174010 2573918 12384975 48117216 0007608975 002218654 0001295014 2915837 17132278 5875593

5

3 001002222 001747601 0003524347 1743726 4958652 28437104 000611229 001333912 0001584625 2182345 8417840 38572475 0004159828 001100573 00008643665 2645717 12732712 48125746 0003042587 000911744 00005193771 2996608 17554567 5858146

20

3 0002467123 000450777 00009057722 1827136 4976715 27237794 0001525546 0003313244 00004123536 2171842 8034958 36996065 0001024757 0002690909 00002184094 2625900 12320482 46919096 00007222134 0002298709 00001300434 3182866 17676476 5553634

100

3 00005033155 00008831595 00001777265 1754684 4969205 28319674 00003078470 00006709811 814811119890 minus 05 2179593 8234807 37781405 00002099437 00005397776 4331869119890 minus 05 2571059 12460617 48464926 00001524544 00004400118 2622736119890 minus 05 2886186 16776824 5812800

200

3 00002471209 00004459992 8781071119890 minus 05 1804781 5079098 28142464 0000153431 00003294056 4053465119890 minus 05 214693 8126519 37851815 00001031339 00002670593 2214463119890 minus 05 2589443 12059777 46572876 7600503119890 minus 05 00002178877 1308801119890 minus 05 2866754 16647886 5807226

500

3 00001002982 00001764347 3693063119890 minus 05 1759102 4777463 27158544 5943145119890 minus 05 00001320398 1595697119890 minus 05 2221716 8274741 37244825 4208938119890 minus 05 00001076813 88025119890 minus 06 2558395 12233036 47815266 3117683119890 minus 05 9065143119890 minus 05 5230148119890 minus 06 2907654 17332479 5960984

Using (14)

119861 = 1 minus1

119899

119899

sum

119894=1

119861 (119901119894) (16)

Var (119861) = var[1 minus 1

119899

119899

sum

119894=1

119861 (119901119894)]

=1

1198992var[119899

sum

119894=1

119861 (119901119894)]

=1

119899119864[119861 (119901)]

2

minus 1198682

2 from (13)

=1

119899[int

1

0

(119861 (119901))2

119889119901 minus (1 minus 119861)2

] from (10)

(17)

Algorithm for computing simple random estimate for Bon-ferroni index is given as follows

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)

(ii) Compute 119901119894= 119883119894 119894 = 1 119873

(iii) Compute 119861(119901119894) 119894 = 1 119873

(iv) Compute 119861 = 1 minus (1119899)sum119899119894=1

119861(119901119894)

ISRN Probability and Statistics 5

Table 8 Relative efficiency of Gini index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002332234 004529824 000589787 1942268 7680441 39543674 001381014 003470658 0002580012 2513122 13452100 53527435 0009193211 002779215 0001352259 3023116 20552387 67984106 000659181 002279101 00008011307 3457473 28448554 8228133

5

3 0009292055 001775528 0002399531 1910802 7399479 38724464 0005570443 001392200 0001050541 2499263 13252220 53024525 0003769394 001075287 00005584852 2852678 19253635 67493186 000268615 0009290222 00003206307 3458554 28974836 8377707

20

3 0002275199 0004430962 00006094044 1947505 7270971 37334804 0001390079 0003490160 00002649288 2510764 13173955 52469915 00009236206 0002648793 00001380824 2867836 19182698 66889096 00006634941 0002265676 8109345119890 minus 05 3414765 27939075 8181846

100

3 0000461317 000090545 00001197988 1962750 7558089 38507654 00002732137 00006699247 5275955119890 minus 05 2452017 12697695 51784695 00001887224 00005433437 2791159119890 minus 05 2879063 19466598 67614356 00001309059 00004580686 1628860119890 minus 05 3499220 28122036 8036658

200

3 00002253251 00004546124 6028802119890 minus 05 2017584 7540676 37374774 00001392043 00003422813 2601848119890 minus 05 2458842 13155315 53502095 9250598119890 minus 05 00002770614 1426628119890 minus 05 2995065 19420718 64842406 6674519119890 minus 05 00002295553 7895343119890 minus 06 3439278 29074772 8453742

500

3 9473933119890 minus 05 00001802806 2443929119890 minus 05 1902912 7376671 38765174 5606348119890 minus 05 00001383186 1074361119890 minus 05 2467178 12874499 52183095 3699474119890 minus 05 00001102943 5493003119890 minus 06 2981350 20079053 67348846 2701999119890 minus 05 9018714119890 minus 05 3301841119890 minus 06 3337793 27314198 8183310

213 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus int

1

0

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901

= minus 119864[AL (119901)119891 (119901)

]

= minus 1198683

(18)

where

AL (119901) = minus120583

2+ int

1

0

119909119865 (119909) 119889119865 (119909) (19)

is the Absolute Lorenz curve defined at 119901 ordinates [6]

1198683= 119864[

AL (119901)119891 (119901)

] (20)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198683= 119864 [AL (119901)] (21)

An unbiased estimator of 119868 is sample mean

1198683=1

119899

119899

sum

119894=1

AL (119901119894) (22)

where 119899 is the sample sizeTherefore unbiased estimator of 119860 is

119860 = minus1198683

(23)

using (22)

119860 = minus1

119899

119899

sum

119894=1

AL (119901119894) (24)

Var (119860) = var[minus1119899

119899

sum

119894=1

AL (119901119894)]

=1

1198992var[119899

sum

119894=1

AL (119901119894)]

=1

119899119864[AL (119901)]2 minus 1198682

3 from (21)

=1

119899[int

1

0

(AL (119901))2119889119901 minus 1198602] from (18)

(25)

Algorithm for computing simple random estimate for Abso-lute Lorenz index is given as follows

6 ISRN Probability and Statistics

Table 9 Relative efficiency of Gini index for power (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 003002883 005904002 0006465061 1966111 9132168 46447874 001837878 004518542 0002769992 2458565 16312473 66349585 001202134 003688425 0001390412 3068232 26527569 86458836 0008600471 002950916 0000838656 3431109 35186250 10255064

5

3 001183113 002385245 000263158 2016075 9063927 44958284 0007109205 001767874 0001112017 2486739 15897904 63930725 0004782672 001397249 00005790662 2921482 24129348 82592846 0003415356 001160659 00003343554 3398354 34713332 10214748

20

3 0003007737 0005933085 00006660064 1972607 8908450 45160784 0001856484 0004438954 00002774993 2391054 15996271 66900495 0001176903 0003597289 00001420534 3056573 25323498 82849346 00008649264 000293501 8396861119890 minus 05 3393364 34953657 10300592

100

3 00006040928 0001167984 00001285668 1933451 9084647 46986694 00003599259 00008980578 5426289119890 minus 05 2495118 16550128 66330035 00002358945 00007081798 2842227119890 minus 05 3002103 24916370 82996366 0000173568 00005906876 1640179119890 minus 05 3403206 36013606 10582260

200

3 00002970297 00005835807 6760029119890 minus 05 1964722 8632814 43939124 00001840207 00004393907 2766382119890 minus 05 2387724 15883226 66520355 00001199216 00003552514 1421686119890 minus 05 2962364 24988035 84351686 884773119890 minus 05 00002953183 821482119890 minus 06 3337786 35949455 10770449

500

3 00001217598 00002407522 2666487119890 minus 05 1977272 9028816 45663004 7261201119890 minus 05 00001784493 1124082119890 minus 05 2457572 15875114 64596725 48425119890 minus 05 00001436364 5589331119890 minus 06 2966163 25698317 86638286 3399731119890 minus 05 00001184066 3256773119890 minus 06 3482823 36357032 10438956

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)(ii) Compute 119901

119894= 119883119894 119894 = 1 119873

(iii) Compute AL(119901119894) 119894 = 1 119873

(iv) Compute 119860 = minus(1119899)sum119899

119894=1AL(119901119894)

Simple random sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in the Table 1

22 Ranked Set Sampling Procedure General ranked setsam-pling procedure involves the following steps [6]

(i) Select a simple random sample of 119898 units fromthe population and subject it to ordering on attribute ofinterestvia some ranking process that is

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

(26)

This judgment ranking can result from a variety ofmechanisms including expertrsquos opinion visual comparisonsor the use of easy-to-obtain auxiliary variables but it cannotinvolve actual measurements of the attribute of interest onsample units

Once this judgement ranking of 119898 units in our initialrandom sample has been accomplished the item judged to besmallest is included as the first item in the ranked set sampleand the attribute of interest is formally measured on this unit

and remaining (119898 minus 1) unmeasured units in the first randomsample are not considered further

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

997904rArr 119883[1]11

(27)

This measurement is denoted by 119883[1]11

the smallestorderstatistics or the smallest judgement order item It mayor may not actually have the smallest attribute measurementamong the 119898 sampled units Note that 119883

(2)11lt sdot sdot sdot lt 119883

(119898)11

are not considered further in the selection of our ranked setsampleThe sole purpose of (119898minus1) units is to help in selectingan item formeasurement that represents the smaller attributevalues in the population

(ii) Following selection of 119883[1]11

a second independentrandom sample of size 119898 is selected from the populationand judgement ranking without formal measurement on theattribute of interest is made and the second smallest item of119898 units in second random sample is selected and is includedin the ranked set sample for measurement of attribute ofinterest that is

119883(1)21

lt sdot sdot sdot lt 119883(119898)21

997904rArr 119883[2]21

(28)

The second measured observation is denoted by119883[2]21

(iii) Similarly from a third independent random sam-

ple we select the unit judgement ranked to be the thirdsmallest119883

[3]31 formeasurement and inclusion in the ranked-

set sample that is

119883(1)31

lt sdot sdot sdot lt 119883(119898)31

997904rArr 119883[3]31

(29)

ISRN Probability and Statistics 7

Table 10 Relative efficiency of Bonferroni index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 006202481 01235527 001377850 1991989 8967065 45015654 00381469 009533465 0005916627 2499146 16113007 64474075 002535770 007558111 0002995199 2980598 25234086 84661156 001791203 006181676 0001748279 3451131 35358636 10245521

5

3 002490822 005036193 0005549141 20219 9075626 44886624 001489808 003685433 0002358322 2473765 15627353 63172375 000982462 003053955 0001206127 3108471 25320344 81455936 0007220886 002487922 00006952421 3445452 35784973 10386146

20

3 0006252341 001245597 0001364699 1992209 9127265 45814804 0003685856 0009230949 00005877362 2504425 15705939 62712765 0002550649 0007501943 00002866664 2941190 26169593 88976216 0001808095 000619635 00001755058 3427005 35305671 10302195

100

3 0001249233 0002446855 00002789078 1958686 8772989 44790184 00007403018 0001914464 00001151496 2586059 16625885 64290445 0000504336 0001465721 601177119890 minus 05 2906238 24380856 83891436 00003529966 0001248915 3414360119890 minus 05 3538037 36578305 10338588

200

3 00006307183 0001248089 00001376957 1978838 9064110 45805234 00003722346 00009302265 5962156119890 minus 05 2499033 15602183 62432895 00002493340 00007560325 2945604119890 minus 05 3032207 25666468 84646146 00001767726 0000618749 1745216119890 minus 05 3500253 35454007 10128981

500

3 00002467436 00005050265 5529099119890 minus 05 2046767 9133975 44626374 00001511701 00003785248 2356102119890 minus 05 2503966 16065722 64161105 998555119890 minus 05 00002998347 1208355119890 minus 05 3002686 24813461 82637556 7085566119890 minus 05 0000252477 6991167119890 minus 06 3563258 36113713 10135026

This process is continued until we have selected the unitjudgement ranked to be largest of119898 units in the119898th randomsample denoted by 119883

[119898]1198981 that is 119883

(1)1198981lt sdot sdot sdot lt 119883

(119898)1198981rArr

119883[119898]1198981

for inclusion in our ranked set sampleThis entire process is referred to as a cycle and the number

of observations in each random sample119898 in our example iscalled set size

Thus to complete a single ranked set cycle we needto judge rank 119898 independent random samples of size 119898

involving a total of 1198982 sample units in order to obtain 119898

measured observations These 119898 observations represent abalanced ranked set sample with set size119898

In practice the sample size 119898 is kept small to ease thevisual ranking the RSS literature suggested 119898 = 2 3 4 5 or6 Therefore in order to obtain a ranked set sample with adesired total number of measured observation 119899 = 119903 lowast 119898 werepeat the entire cycle process 119903 independent times yieldingthe data

119883[1]119894119895

lt sdot sdot sdot lt 119883[119898]119894119895

119894 = 1 119898 119895 = 1 119903 (30)

Algorithm for obtaining the RSS estimate of Gini indexvia method of Monte Carlo simulation is given as follows

(a) Generate an RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119871(119901

(119894)119895)

(d) Compute the ranked set sample estimate ofGini indexas

119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871 (119901(119894)119895)

119899 (31)

Algorithm for obtaining the RSS estimate of Bonferroniindex via method of Monte Carlo simulation is given asfollowing

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119861(119901

(119894)119895)

(d) Compute the ranked set sample estimate of Bonfer-roni index as follows

119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861 (119901(119894)119895)

119899 (32)

Algorithm for obtaining the RSS estimate of AbsoluteLorenz index via method of Monte Carlo simulation is givenas follows

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 5

Table 8 Relative efficiency of Gini index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 002332234 004529824 000589787 1942268 7680441 39543674 001381014 003470658 0002580012 2513122 13452100 53527435 0009193211 002779215 0001352259 3023116 20552387 67984106 000659181 002279101 00008011307 3457473 28448554 8228133

5

3 0009292055 001775528 0002399531 1910802 7399479 38724464 0005570443 001392200 0001050541 2499263 13252220 53024525 0003769394 001075287 00005584852 2852678 19253635 67493186 000268615 0009290222 00003206307 3458554 28974836 8377707

20

3 0002275199 0004430962 00006094044 1947505 7270971 37334804 0001390079 0003490160 00002649288 2510764 13173955 52469915 00009236206 0002648793 00001380824 2867836 19182698 66889096 00006634941 0002265676 8109345119890 minus 05 3414765 27939075 8181846

100

3 0000461317 000090545 00001197988 1962750 7558089 38507654 00002732137 00006699247 5275955119890 minus 05 2452017 12697695 51784695 00001887224 00005433437 2791159119890 minus 05 2879063 19466598 67614356 00001309059 00004580686 1628860119890 minus 05 3499220 28122036 8036658

200

3 00002253251 00004546124 6028802119890 minus 05 2017584 7540676 37374774 00001392043 00003422813 2601848119890 minus 05 2458842 13155315 53502095 9250598119890 minus 05 00002770614 1426628119890 minus 05 2995065 19420718 64842406 6674519119890 minus 05 00002295553 7895343119890 minus 06 3439278 29074772 8453742

500

3 9473933119890 minus 05 00001802806 2443929119890 minus 05 1902912 7376671 38765174 5606348119890 minus 05 00001383186 1074361119890 minus 05 2467178 12874499 52183095 3699474119890 minus 05 00001102943 5493003119890 minus 06 2981350 20079053 67348846 2701999119890 minus 05 9018714119890 minus 05 3301841119890 minus 06 3337793 27314198 8183310

213 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus int

1

0

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901

= minus 119864[AL (119901)119891 (119901)

]

= minus 1198683

(18)

where

AL (119901) = minus120583

2+ int

1

0

119909119865 (119909) 119889119865 (119909) (19)

is the Absolute Lorenz curve defined at 119901 ordinates [6]

1198683= 119864[

AL (119901)119891 (119901)

] (20)

Since 119901 is distributed uniformly over [0 1] that is 119901 sim

119880(0 1)

997904rArr 1198683= 119864 [AL (119901)] (21)

An unbiased estimator of 119868 is sample mean

1198683=1

119899

119899

sum

119894=1

AL (119901119894) (22)

where 119899 is the sample sizeTherefore unbiased estimator of 119860 is

119860 = minus1198683

(23)

using (22)

119860 = minus1

119899

119899

sum

119894=1

AL (119901119894) (24)

Var (119860) = var[minus1119899

119899

sum

119894=1

AL (119901119894)]

=1

1198992var[119899

sum

119894=1

AL (119901119894)]

=1

119899119864[AL (119901)]2 minus 1198682

3 from (21)

=1

119899[int

1

0

(AL (119901))2119889119901 minus 1198602] from (18)

(25)

Algorithm for computing simple random estimate for Abso-lute Lorenz index is given as follows

6 ISRN Probability and Statistics

Table 9 Relative efficiency of Gini index for power (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 003002883 005904002 0006465061 1966111 9132168 46447874 001837878 004518542 0002769992 2458565 16312473 66349585 001202134 003688425 0001390412 3068232 26527569 86458836 0008600471 002950916 0000838656 3431109 35186250 10255064

5

3 001183113 002385245 000263158 2016075 9063927 44958284 0007109205 001767874 0001112017 2486739 15897904 63930725 0004782672 001397249 00005790662 2921482 24129348 82592846 0003415356 001160659 00003343554 3398354 34713332 10214748

20

3 0003007737 0005933085 00006660064 1972607 8908450 45160784 0001856484 0004438954 00002774993 2391054 15996271 66900495 0001176903 0003597289 00001420534 3056573 25323498 82849346 00008649264 000293501 8396861119890 minus 05 3393364 34953657 10300592

100

3 00006040928 0001167984 00001285668 1933451 9084647 46986694 00003599259 00008980578 5426289119890 minus 05 2495118 16550128 66330035 00002358945 00007081798 2842227119890 minus 05 3002103 24916370 82996366 0000173568 00005906876 1640179119890 minus 05 3403206 36013606 10582260

200

3 00002970297 00005835807 6760029119890 minus 05 1964722 8632814 43939124 00001840207 00004393907 2766382119890 minus 05 2387724 15883226 66520355 00001199216 00003552514 1421686119890 minus 05 2962364 24988035 84351686 884773119890 minus 05 00002953183 821482119890 minus 06 3337786 35949455 10770449

500

3 00001217598 00002407522 2666487119890 minus 05 1977272 9028816 45663004 7261201119890 minus 05 00001784493 1124082119890 minus 05 2457572 15875114 64596725 48425119890 minus 05 00001436364 5589331119890 minus 06 2966163 25698317 86638286 3399731119890 minus 05 00001184066 3256773119890 minus 06 3482823 36357032 10438956

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)(ii) Compute 119901

119894= 119883119894 119894 = 1 119873

(iii) Compute AL(119901119894) 119894 = 1 119873

(iv) Compute 119860 = minus(1119899)sum119899

119894=1AL(119901119894)

Simple random sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in the Table 1

22 Ranked Set Sampling Procedure General ranked setsam-pling procedure involves the following steps [6]

(i) Select a simple random sample of 119898 units fromthe population and subject it to ordering on attribute ofinterestvia some ranking process that is

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

(26)

This judgment ranking can result from a variety ofmechanisms including expertrsquos opinion visual comparisonsor the use of easy-to-obtain auxiliary variables but it cannotinvolve actual measurements of the attribute of interest onsample units

Once this judgement ranking of 119898 units in our initialrandom sample has been accomplished the item judged to besmallest is included as the first item in the ranked set sampleand the attribute of interest is formally measured on this unit

and remaining (119898 minus 1) unmeasured units in the first randomsample are not considered further

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

997904rArr 119883[1]11

(27)

This measurement is denoted by 119883[1]11

the smallestorderstatistics or the smallest judgement order item It mayor may not actually have the smallest attribute measurementamong the 119898 sampled units Note that 119883

(2)11lt sdot sdot sdot lt 119883

(119898)11

are not considered further in the selection of our ranked setsampleThe sole purpose of (119898minus1) units is to help in selectingan item formeasurement that represents the smaller attributevalues in the population

(ii) Following selection of 119883[1]11

a second independentrandom sample of size 119898 is selected from the populationand judgement ranking without formal measurement on theattribute of interest is made and the second smallest item of119898 units in second random sample is selected and is includedin the ranked set sample for measurement of attribute ofinterest that is

119883(1)21

lt sdot sdot sdot lt 119883(119898)21

997904rArr 119883[2]21

(28)

The second measured observation is denoted by119883[2]21

(iii) Similarly from a third independent random sam-

ple we select the unit judgement ranked to be the thirdsmallest119883

[3]31 formeasurement and inclusion in the ranked-

set sample that is

119883(1)31

lt sdot sdot sdot lt 119883(119898)31

997904rArr 119883[3]31

(29)

ISRN Probability and Statistics 7

Table 10 Relative efficiency of Bonferroni index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 006202481 01235527 001377850 1991989 8967065 45015654 00381469 009533465 0005916627 2499146 16113007 64474075 002535770 007558111 0002995199 2980598 25234086 84661156 001791203 006181676 0001748279 3451131 35358636 10245521

5

3 002490822 005036193 0005549141 20219 9075626 44886624 001489808 003685433 0002358322 2473765 15627353 63172375 000982462 003053955 0001206127 3108471 25320344 81455936 0007220886 002487922 00006952421 3445452 35784973 10386146

20

3 0006252341 001245597 0001364699 1992209 9127265 45814804 0003685856 0009230949 00005877362 2504425 15705939 62712765 0002550649 0007501943 00002866664 2941190 26169593 88976216 0001808095 000619635 00001755058 3427005 35305671 10302195

100

3 0001249233 0002446855 00002789078 1958686 8772989 44790184 00007403018 0001914464 00001151496 2586059 16625885 64290445 0000504336 0001465721 601177119890 minus 05 2906238 24380856 83891436 00003529966 0001248915 3414360119890 minus 05 3538037 36578305 10338588

200

3 00006307183 0001248089 00001376957 1978838 9064110 45805234 00003722346 00009302265 5962156119890 minus 05 2499033 15602183 62432895 00002493340 00007560325 2945604119890 minus 05 3032207 25666468 84646146 00001767726 0000618749 1745216119890 minus 05 3500253 35454007 10128981

500

3 00002467436 00005050265 5529099119890 minus 05 2046767 9133975 44626374 00001511701 00003785248 2356102119890 minus 05 2503966 16065722 64161105 998555119890 minus 05 00002998347 1208355119890 minus 05 3002686 24813461 82637556 7085566119890 minus 05 0000252477 6991167119890 minus 06 3563258 36113713 10135026

This process is continued until we have selected the unitjudgement ranked to be largest of119898 units in the119898th randomsample denoted by 119883

[119898]1198981 that is 119883

(1)1198981lt sdot sdot sdot lt 119883

(119898)1198981rArr

119883[119898]1198981

for inclusion in our ranked set sampleThis entire process is referred to as a cycle and the number

of observations in each random sample119898 in our example iscalled set size

Thus to complete a single ranked set cycle we needto judge rank 119898 independent random samples of size 119898

involving a total of 1198982 sample units in order to obtain 119898

measured observations These 119898 observations represent abalanced ranked set sample with set size119898

In practice the sample size 119898 is kept small to ease thevisual ranking the RSS literature suggested 119898 = 2 3 4 5 or6 Therefore in order to obtain a ranked set sample with adesired total number of measured observation 119899 = 119903 lowast 119898 werepeat the entire cycle process 119903 independent times yieldingthe data

119883[1]119894119895

lt sdot sdot sdot lt 119883[119898]119894119895

119894 = 1 119898 119895 = 1 119903 (30)

Algorithm for obtaining the RSS estimate of Gini indexvia method of Monte Carlo simulation is given as follows

(a) Generate an RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119871(119901

(119894)119895)

(d) Compute the ranked set sample estimate ofGini indexas

119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871 (119901(119894)119895)

119899 (31)

Algorithm for obtaining the RSS estimate of Bonferroniindex via method of Monte Carlo simulation is given asfollowing

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119861(119901

(119894)119895)

(d) Compute the ranked set sample estimate of Bonfer-roni index as follows

119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861 (119901(119894)119895)

119899 (32)

Algorithm for obtaining the RSS estimate of AbsoluteLorenz index via method of Monte Carlo simulation is givenas follows

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Stochastic AnalysisInternational Journal of

6 ISRN Probability and Statistics

Table 9 Relative efficiency of Gini index for power (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 003002883 005904002 0006465061 1966111 9132168 46447874 001837878 004518542 0002769992 2458565 16312473 66349585 001202134 003688425 0001390412 3068232 26527569 86458836 0008600471 002950916 0000838656 3431109 35186250 10255064

5

3 001183113 002385245 000263158 2016075 9063927 44958284 0007109205 001767874 0001112017 2486739 15897904 63930725 0004782672 001397249 00005790662 2921482 24129348 82592846 0003415356 001160659 00003343554 3398354 34713332 10214748

20

3 0003007737 0005933085 00006660064 1972607 8908450 45160784 0001856484 0004438954 00002774993 2391054 15996271 66900495 0001176903 0003597289 00001420534 3056573 25323498 82849346 00008649264 000293501 8396861119890 minus 05 3393364 34953657 10300592

100

3 00006040928 0001167984 00001285668 1933451 9084647 46986694 00003599259 00008980578 5426289119890 minus 05 2495118 16550128 66330035 00002358945 00007081798 2842227119890 minus 05 3002103 24916370 82996366 0000173568 00005906876 1640179119890 minus 05 3403206 36013606 10582260

200

3 00002970297 00005835807 6760029119890 minus 05 1964722 8632814 43939124 00001840207 00004393907 2766382119890 minus 05 2387724 15883226 66520355 00001199216 00003552514 1421686119890 minus 05 2962364 24988035 84351686 884773119890 minus 05 00002953183 821482119890 minus 06 3337786 35949455 10770449

500

3 00001217598 00002407522 2666487119890 minus 05 1977272 9028816 45663004 7261201119890 minus 05 00001784493 1124082119890 minus 05 2457572 15875114 64596725 48425119890 minus 05 00001436364 5589331119890 minus 06 2966163 25698317 86638286 3399731119890 minus 05 00001184066 3256773119890 minus 06 3482823 36357032 10438956

(i) Generate a sequence 119883119894119873

119894=1of 119873 random numbers

from 119880(0 1)(ii) Compute 119901

119894= 119883119894 119894 = 1 119873

(iii) Compute AL(119901119894) 119894 = 1 119873

(iv) Compute 119860 = minus(1119899)sum119899

119894=1AL(119901119894)

Simple random sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in the Table 1

22 Ranked Set Sampling Procedure General ranked setsam-pling procedure involves the following steps [6]

(i) Select a simple random sample of 119898 units fromthe population and subject it to ordering on attribute ofinterestvia some ranking process that is

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

(26)

This judgment ranking can result from a variety ofmechanisms including expertrsquos opinion visual comparisonsor the use of easy-to-obtain auxiliary variables but it cannotinvolve actual measurements of the attribute of interest onsample units

Once this judgement ranking of 119898 units in our initialrandom sample has been accomplished the item judged to besmallest is included as the first item in the ranked set sampleand the attribute of interest is formally measured on this unit

and remaining (119898 minus 1) unmeasured units in the first randomsample are not considered further

119883(1)11

lt sdot sdot sdot lt 119883(119898)11

997904rArr 119883[1]11

(27)

This measurement is denoted by 119883[1]11

the smallestorderstatistics or the smallest judgement order item It mayor may not actually have the smallest attribute measurementamong the 119898 sampled units Note that 119883

(2)11lt sdot sdot sdot lt 119883

(119898)11

are not considered further in the selection of our ranked setsampleThe sole purpose of (119898minus1) units is to help in selectingan item formeasurement that represents the smaller attributevalues in the population

(ii) Following selection of 119883[1]11

a second independentrandom sample of size 119898 is selected from the populationand judgement ranking without formal measurement on theattribute of interest is made and the second smallest item of119898 units in second random sample is selected and is includedin the ranked set sample for measurement of attribute ofinterest that is

119883(1)21

lt sdot sdot sdot lt 119883(119898)21

997904rArr 119883[2]21

(28)

The second measured observation is denoted by119883[2]21

(iii) Similarly from a third independent random sam-

ple we select the unit judgement ranked to be the thirdsmallest119883

[3]31 formeasurement and inclusion in the ranked-

set sample that is

119883(1)31

lt sdot sdot sdot lt 119883(119898)31

997904rArr 119883[3]31

(29)

ISRN Probability and Statistics 7

Table 10 Relative efficiency of Bonferroni index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 006202481 01235527 001377850 1991989 8967065 45015654 00381469 009533465 0005916627 2499146 16113007 64474075 002535770 007558111 0002995199 2980598 25234086 84661156 001791203 006181676 0001748279 3451131 35358636 10245521

5

3 002490822 005036193 0005549141 20219 9075626 44886624 001489808 003685433 0002358322 2473765 15627353 63172375 000982462 003053955 0001206127 3108471 25320344 81455936 0007220886 002487922 00006952421 3445452 35784973 10386146

20

3 0006252341 001245597 0001364699 1992209 9127265 45814804 0003685856 0009230949 00005877362 2504425 15705939 62712765 0002550649 0007501943 00002866664 2941190 26169593 88976216 0001808095 000619635 00001755058 3427005 35305671 10302195

100

3 0001249233 0002446855 00002789078 1958686 8772989 44790184 00007403018 0001914464 00001151496 2586059 16625885 64290445 0000504336 0001465721 601177119890 minus 05 2906238 24380856 83891436 00003529966 0001248915 3414360119890 minus 05 3538037 36578305 10338588

200

3 00006307183 0001248089 00001376957 1978838 9064110 45805234 00003722346 00009302265 5962156119890 minus 05 2499033 15602183 62432895 00002493340 00007560325 2945604119890 minus 05 3032207 25666468 84646146 00001767726 0000618749 1745216119890 minus 05 3500253 35454007 10128981

500

3 00002467436 00005050265 5529099119890 minus 05 2046767 9133975 44626374 00001511701 00003785248 2356102119890 minus 05 2503966 16065722 64161105 998555119890 minus 05 00002998347 1208355119890 minus 05 3002686 24813461 82637556 7085566119890 minus 05 0000252477 6991167119890 minus 06 3563258 36113713 10135026

This process is continued until we have selected the unitjudgement ranked to be largest of119898 units in the119898th randomsample denoted by 119883

[119898]1198981 that is 119883

(1)1198981lt sdot sdot sdot lt 119883

(119898)1198981rArr

119883[119898]1198981

for inclusion in our ranked set sampleThis entire process is referred to as a cycle and the number

of observations in each random sample119898 in our example iscalled set size

Thus to complete a single ranked set cycle we needto judge rank 119898 independent random samples of size 119898

involving a total of 1198982 sample units in order to obtain 119898

measured observations These 119898 observations represent abalanced ranked set sample with set size119898

In practice the sample size 119898 is kept small to ease thevisual ranking the RSS literature suggested 119898 = 2 3 4 5 or6 Therefore in order to obtain a ranked set sample with adesired total number of measured observation 119899 = 119903 lowast 119898 werepeat the entire cycle process 119903 independent times yieldingthe data

119883[1]119894119895

lt sdot sdot sdot lt 119883[119898]119894119895

119894 = 1 119898 119895 = 1 119903 (30)

Algorithm for obtaining the RSS estimate of Gini indexvia method of Monte Carlo simulation is given as follows

(a) Generate an RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119871(119901

(119894)119895)

(d) Compute the ranked set sample estimate ofGini indexas

119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871 (119901(119894)119895)

119899 (31)

Algorithm for obtaining the RSS estimate of Bonferroniindex via method of Monte Carlo simulation is given asfollowing

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119861(119901

(119894)119895)

(d) Compute the ranked set sample estimate of Bonfer-roni index as follows

119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861 (119901(119894)119895)

119899 (32)

Algorithm for obtaining the RSS estimate of AbsoluteLorenz index via method of Monte Carlo simulation is givenas follows

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 7

Table 10 Relative efficiency of Bonferroni index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 006202481 01235527 001377850 1991989 8967065 45015654 00381469 009533465 0005916627 2499146 16113007 64474075 002535770 007558111 0002995199 2980598 25234086 84661156 001791203 006181676 0001748279 3451131 35358636 10245521

5

3 002490822 005036193 0005549141 20219 9075626 44886624 001489808 003685433 0002358322 2473765 15627353 63172375 000982462 003053955 0001206127 3108471 25320344 81455936 0007220886 002487922 00006952421 3445452 35784973 10386146

20

3 0006252341 001245597 0001364699 1992209 9127265 45814804 0003685856 0009230949 00005877362 2504425 15705939 62712765 0002550649 0007501943 00002866664 2941190 26169593 88976216 0001808095 000619635 00001755058 3427005 35305671 10302195

100

3 0001249233 0002446855 00002789078 1958686 8772989 44790184 00007403018 0001914464 00001151496 2586059 16625885 64290445 0000504336 0001465721 601177119890 minus 05 2906238 24380856 83891436 00003529966 0001248915 3414360119890 minus 05 3538037 36578305 10338588

200

3 00006307183 0001248089 00001376957 1978838 9064110 45805234 00003722346 00009302265 5962156119890 minus 05 2499033 15602183 62432895 00002493340 00007560325 2945604119890 minus 05 3032207 25666468 84646146 00001767726 0000618749 1745216119890 minus 05 3500253 35454007 10128981

500

3 00002467436 00005050265 5529099119890 minus 05 2046767 9133975 44626374 00001511701 00003785248 2356102119890 minus 05 2503966 16065722 64161105 998555119890 minus 05 00002998347 1208355119890 minus 05 3002686 24813461 82637556 7085566119890 minus 05 0000252477 6991167119890 minus 06 3563258 36113713 10135026

This process is continued until we have selected the unitjudgement ranked to be largest of119898 units in the119898th randomsample denoted by 119883

[119898]1198981 that is 119883

(1)1198981lt sdot sdot sdot lt 119883

(119898)1198981rArr

119883[119898]1198981

for inclusion in our ranked set sampleThis entire process is referred to as a cycle and the number

of observations in each random sample119898 in our example iscalled set size

Thus to complete a single ranked set cycle we needto judge rank 119898 independent random samples of size 119898

involving a total of 1198982 sample units in order to obtain 119898

measured observations These 119898 observations represent abalanced ranked set sample with set size119898

In practice the sample size 119898 is kept small to ease thevisual ranking the RSS literature suggested 119898 = 2 3 4 5 or6 Therefore in order to obtain a ranked set sample with adesired total number of measured observation 119899 = 119903 lowast 119898 werepeat the entire cycle process 119903 independent times yieldingthe data

119883[1]119894119895

lt sdot sdot sdot lt 119883[119898]119894119895

119894 = 1 119898 119895 = 1 119903 (30)

Algorithm for obtaining the RSS estimate of Gini indexvia method of Monte Carlo simulation is given as follows

(a) Generate an RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119871(119901

(119894)119895)

(d) Compute the ranked set sample estimate ofGini indexas

119866 = 1 minus 2

sum119903

119895=1sum119898

119894=1119871 (119901(119894)119895)

119899 (31)

Algorithm for obtaining the RSS estimate of Bonferroniindex via method of Monte Carlo simulation is given asfollowing

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903(c) Compute 119861(119901

(119894)119895)

(d) Compute the ranked set sample estimate of Bonfer-roni index as follows

119861 = 1 minus

sum119903

119895=1sum119898

119894=1119861 (119901(119894)119895)

119899 (32)

Algorithm for obtaining the RSS estimate of AbsoluteLorenz index via method of Monte Carlo simulation is givenas follows

(a) Generate a RSS 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903 ofsize 119899 = 119898 times 119903 from 119880(0 1)

(b) Assign 119901(119894)119895

= 119880[119898]119894119895

119894 = 1 119898 119895 = 1 119903

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 ISRN Probability and Statistics

Table 11 Relative efficiency of Bonferroni index for exponential distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 0005826023 001085295 0001687301 186284 6432136 34528654 0003477376 0008182712 00007416163 2353128 11033619 46889155 0002229820 0006484168 00003876997 2907933 16724718 57514106 0001690472 0005454195 00002310937 3226432 23601660 7315093

5

3 0002319801 0004423554 00006583515 1906868 6719137 35236514 0001337874 0003238717 00002952573 2420794 10969134 45312145 00009042652 0002668140 00001607564 2950617 16597411 56250656 00006640723 0002151227 939234119890 minus 05 3239447 22904058 7070361

20

3 00005728049 0001073308 00001686894 1873776 6362629 33956194 00003477821 00008003468 7638552119890 minus 05 2301288 10477729 45529855 00002317084 00006640685 4120043119890 minus 05 2865966 16117999 56239326 00001682733 00005685107 2490111119890 minus 05 3378496 22830737 6757663

100

3 00001145552 00002185333 3457197119890 minus 05 1907668 6321112 33135284 7047152119890 minus 05 00001665346 1634645119890 minus 05 2363148 10187814 43111215 476686119890 minus 05 00001299972 9897842119890 minus 06 2727103 13133893 48160606 3368466119890 minus 05 00001080593 6489773119890 minus 06 320796 16650706 5190422

200

3 5866765119890 minus 05 00001113661 1836847119890 minus 05 1898254 6062895 31939324 3637083119890 minus 05 833706119890 minus 05 9291632119890 minus 06 2292238 8972654 39143645 2444185119890 minus 05 6776085119890 minus 05 5651263119890 minus 06 2772329 11990390 43250246 1846277119890 minus 05 5761897119890 minus 05 405105119890 minus 06 3120819 14223219 4557527

500

3 2478413119890 minus 05 4502048119890 minus 05 8473576119890 minus 06 1816505 5313044 29248734 1551794119890 minus 05 3491700119890 minus 05 4598167119890 minus 06 2250106 7593678 33748105 1109970119890 minus 05 2797701119890 minus 05 3230694119890 minus 06 2520520 8659752 34357016 8396832119890 minus 06 2362577119890 minus 05 2635701119890 minus 06 2813653 8963752 3185806

(c) Compute AL(119901(119894)119895)

(d) Compute the ranked set sample estimate of AbsoluteLorenz index as

119860 = minus

sum119903

119895=1sum119898

119894=1AL (119901

(119894)119895)

119899 (33)

Ranked set sampling estimates of Gini Bonferroni andAbsolute Lorenz indices using Monte Carlo approach aregiven in Table 2

23 Systematic Sampling Procedure Systematic samplingprocedure is particular case of stratified sampling procedurewhere the stratas are of equal size

231 Gini Index Gini index is defined as

119866 = 1 minus 2int

1

0

119871 (119901) 119889119901

= 1 minus 21198681

(34)

Its unbiased estimator in case of stratified sampling is givenas

997904rArr 119866 = 1 minus 21198681 (35)

where

1198681=int

1

0

119871 (119901) 119889119901 = int119863

119871 (119901) 119889119901

=int119863

119871 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(36)

where

119892 (119901) =119871 (119901)

119891 (119901) (37)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198681119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198681= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198681119894

(38)

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 9

Table 12 Relative efficiency of Bonferroni index for pareto (120572 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00004945458 00009228851 00001830762 1866127 5040989 27013114 00003013004 00007029501 8527298119890 minus 05 2333054 8243527 35333635 00002011407 00005511385 4790515119890 minus 05 2740064 11504786 41987286 00001507043 00004636178 2906871119890 minus 05 3076342 15949032 5184417

5

3 00001979686 00003752962 7295216119890 minus 05 1895736 5144415 27136774 00001211198 00002751505 3327257119890 minus 05 2271723 8269590 36402305 827896119890 minus 05 00002221785 1902978119890 minus 05 2683652 11675306 43505286 593931119890 minus 05 00001855068 1188047119890 minus 05 3123373 15614433 4999221

20

3 4902086119890 minus 05 9181063119890 minus 05 1821531119890 minus 05 1872889 5040300 26911904 3027866119890 minus 05 7056706119890 minus 05 884171119890 minus 06 2330587 7981155 34245255 2076647119890 minus 05 5352124119890 minus 05 4825012119890 minus 06 2577291 11092457 43039216 1512937119890 minus 05 4587691119890 minus 05 2918024119890 minus 06 3032308 15721910 5184800

100

3 994706119890 minus 06 1860485119890 minus 05 3674845119890 minus 06 1870386 5062758 27067974 6097801119890 minus 06 1383726119890 minus 05 1745582119890 minus 06 2269221 7927018 34932775 3963538119890 minus 06 1106979119890 minus 05 9846638119890 minus 07 2792905 11242203 40252706 2950961119890 minus 06 9162954119890 minus 06 6031919119890 minus 07 3105075 15190778 4892242

200

3 4966537119890 minus 06 9244004119890 minus 06 1813586119890 minus 06 1861257 5097086 27385184 2995594119890 minus 06 6901277119890 minus 06 8687236119890 minus 07 2303809 7944157 34482715 2069782119890 minus 06 5520984119890 minus 06 4823448119890 minus 07 2667423 11446136 42910846 1466736119890 minus 06 4654902119890 minus 06 2952955119890 minus 07 3173647 15763539 4967011

500

3 1986694119890 minus 06 373913119890 minus 06 7290523119890 minus 07 1882087 5128754 27250364 1206677119890 minus 06 2715055119890 minus 06 3486712119890 minus 07 2250026 7786863 34607885 8166757119890 minus 07 2197725119890 minus 06 1930648119890 minus 07 2691062 11383354 42300606 5903014119890 minus 07 1834876119890 minus 06 1190148119890 minus 07 3108371 15417209 4959899

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198681119894=int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(39)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (40)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198681119894can be estimated by

1198681119894=

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (41)

Therefore integral 1198681is estimated by

1198681=

119898

sum

119894=1

1198681119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (42)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119866 = 1 minus 21198681

119866 = 1 minus 2

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198681) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(43)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

1119894

1198752

119894

Var (119866) = Var (1 minus 21198681) = 4Var (119868

1) = 4 sdot

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(44)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Gini index as

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (45)

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 ISRN Probability and Statistics

Table 13 Relative efficiency of absolute Lorenz index for pareto (120572 = 3 1199090= 200) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 2070694 2860494 1057485 1381418 2704997 19581314 135872 2177554 5390407 1602651 4039684 25206265 9815137 1737329 306463 1770051 5668968 32027156 7333079 143627 1935963 1958618 7418892 3787820

5

3 8490195 1132587 4191259 1333994 2702260 20256914 5523156 8495127 2138295 1538093 3972851 25829725 3839579 6901874 121959 179756 5659176 31482546 2848677 5687296 07867597 1996469 7228759 3620771

20

3 2050789 2857925 1037954 1393574 2753422 19758004 1377661 2149993 0521389 1560612 4123587 26422905 09706388 1714577 0301321 1766442 5690201 32212786 07303821 1413525 01940954 1935322 7282630 3763006

100

3 04120384 05679051 02109544 1378282 2692075 19532114 0270179 04210431 01080168 1558386 3897941 25012685 01956007 03382184 006211348 1729127 5445169 31490866 01465505 02881033 003872126 1965897 7440442 3784755

200

3 02098861 02912766 01067612 1387785 2728300 19659404 01389657 02102593 005198104 1513031 4044923 26733925 01003489 01699745 003067446 1693835 5541239 32714156 007082068 01436694 001940810 2028637 7402548 3649027

500

3 008326179 01144187 004192018 1374205 2729442 19861984 00548825 008290751 002124204 1510636 3902992 25836745 003854217 00686116 001242014 1780170 5524221 31031996 002853415 005816632 0007684862 2038481 7568948 3713033

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119866 = 1 minus2

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (46)

232 Bonferroni Index Similarly Bonferroni index isdefined as

119861 = 1 minus int

1

0

119861 (119901) 119889119901

= 1 minus 1198682

(47)

Its stratified estimate is given as

997904rArr 119861 = 1 minus 1198682 (48)

where

1198682=int

1

0

119861 (119901) 119889119901 = int119863

119861 (119901) 119889119901

=int119863

119861 (119901)

119891 (119901)sdot 119891 (119901) 119889119901 = int

119863

119892 (119901) sdot 119891 (119901) 119889119901

(49)

where

119892 (119901) =119861 (119901)

119891 (119901) (50)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is 119863 = ⋃119898

119894=1119863119894and 119863

119896cap 119863119895= 0 119896 = 119895 where

0 is null setDefine

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198682= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198682119894

(51)

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 11

Table 14 Relative efficiency of absolute Lorenz index for rectangular (120579 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00018881 0002099699 000105751 111207 1985512 17854204 0001239755 0001571081 00004786396 1267251 3282388 25901645 00008749366 0001232462 00002404251 1408630 5126179 36391236 00006767988 0001043956 00001412633 1542491 7390143 4791045

5

3 00007426104 0000812753 00004213969 1094454 1928711 17622594 00005023406 00006275718 00001826594 1249295 3435749 27501495 00003583338 00005023294 973136119890 minus 05 1401848 5161965 36822586 00002722183 00004232076 5552897119890 minus 05 1554662 7621384 4902275

20

3 00001900359 00002072712 00001066924 1090695 1942699 17811574 00001270530 00001569948 4505728119890 minus 05 1235664 3484338 28198115 921853119890 minus 05 00001217164 2413549119890 minus 05 1320345 5043047 38194926 6948416119890 minus 05 00001039845 1432389119890 minus 05 1496521 7259515 4850928

100

3 3739423119890 minus 05 4054800119890 minus 05 2125268119890 minus 05 1084339 1907901 17595074 2485619119890 minus 05 3087114119890 minus 05 9368584119890 minus 06 1241990 3295177 26531435 1785318119890 minus 05 2452328119890 minus 05 4819807119890 minus 06 1373609 5088021 37041286 1317393119890 minus 05 2022185119890 minus 05 2820976119890 minus 06 1534990 7168388 4669990

200

3 1865214119890 minus 05 20668119890 minus 05 1051127119890 minus 05 1108077 1966270 17744904 1261365119890 minus 05 1541276119890 minus 05 4629849119890 minus 06 1221911 3328998 27244195 9145499119890 minus 06 1250770119890 minus 05 2469242119890 minus 06 1367635 5065401 37037686 661361119890 minus 06 1047759119890 minus 05 1394760119890 minus 06 1584246 7512110 4741755

500

3 7542962119890 minus 06 8347197119890 minus 06 4224176119890 minus 06 1106621 1976053 17856654 5027219119890 minus 06 6440567119890 minus 06 1884210119890 minus 06 1281139 3418179 26680785 3551602119890 minus 06 4918697119890 minus 06 9686214119890 minus 07 1384923 5078039 36666576 265354119890 minus 06 4210024119890 minus 06 5679265119890 minus 07 1586569 7412973 4672330

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198682119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(52)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (53)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198682119894can be estimated by

1198682119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (54)

Therefore integral 1198682is estimated by

1198682=

119898

sum

119894=1

1198682119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (55)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119861 = 1 minus 1198682

119861 = 1 minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198682) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(56)

where

1205902

119894= Var [119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

2119894

1198752

119894

Var (119866) = Var (1 minus 1198682) = Var (119868

2) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(57)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Bonferroni index as

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (58)

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 ISRN Probability and Statistics

Table 15 Relative efficiency of absolute Lorenz index for exponential (120582 = 3) distribution

119903 119898 MSE (RSS) MSE (SRS) MSE (SYS) 119864(1) 119864(2) 119864(3)

2

3 00001772438 00002139371 998993119890 minus 05 1207022 2141528 17742254 00001160937 0001597166 4811047119890 minus 05 1375756 3319789 24130655 8333168119890 minus 05 00001291083 2622109119890 minus 05 1549330 4923834 31780406 631974119890 minus 05 00001062572 1622325119890 minus 05 1681354 6549686 3895483

5

3 701547119890 minus 05 8522752119890 minus 05 3934262119890 minus 05 1214851 2166290 17831734 4750068119890 minus 05 6440414119890 minus 05 1899583119890 minus 05 1355857 3390436 25005855 3310539119890 minus 05 5144982119890 minus 05 1046542119890 minus 05 1554122 4916173 31633126 2532269119890 minus 05 4341858119890 minus 05 632867119890 minus 06 1714612 6860617 4001266

20

3 1813884119890 minus 05 2141139119890 minus 05 1017992119890 minus 05 1180417 2103296 17818254 1161498119890 minus 05 1671353119890 minus 05 4712122119890 minus 06 1438963 3546922 24649155 8348938119890 minus 06 1282813119890 minus 05 2611685119890 minus 06 1536498 4911821 31967636 6300591119890 minus 06 1076669119890 minus 05 1601593119890 minus 06 1708839 6722488 3933953

100

3 3576201119890 minus 06 4282342119890 minus 06 1949306119890 minus 06 1197455 2196855 18346024 2350533119890 minus 06 3236420119890 minus 06 9673106119890 minus 07 1376887 3345792 24299675 1667317119890 minus 06 2622885119890 minus 06 5363255119890 minus 07 1573117 4890472 31087786 1220823119890 minus 06 2203291119890 minus 06 3159623119890 minus 07 1804759 6973272 3863825

200

3 1776957119890 minus 06 2109820119890 minus 06 1023774119890 minus 06 1187322 2060826 17356934 1175403119890 minus 06 1581642119890 minus 06 478013119890 minus 07 1345617 3308784 24589355 8277293119890 minus 07 1306533119890 minus 06 2650969119890 minus 07 1578454 4928511 31223656 6383724119890 minus 07 1065240119890 minus 06 1576827119890 minus 07 1668682 6755592 4048462

500

3 7048948119890 minus 07 8251703119890 minus 07 4060795119890 minus 07 1170629 2032041 17358544 4695757119890 minus 07 6461275119890 minus 07 1943934119890 minus 07 1375981 3323814 24155955 3350298119890 minus 07 5157602119890 minus 07 1047704119890 minus 07 1539446 4922766 31977526 2468869119890 minus 07 4345264119890 minus 07 6523285119890 minus 08 1760022 6661159 3784702

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896 119896 = 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119861 = 1 minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (59)

233 Absolute Lorenz Index Absolute Lorenz index isdefined as

119860 = minus int

1

0

AL (119901) 119889119901

= minus 1198683

(60)

Its stratified estimate is given as

997904rArr 119860 = minus1198683 (61)

where

1198683=int

1

0

AL (119901) 119889119901 = int119863

AL (119901) 119889119901

=int119863

AL (119901)119891 (119901)

sdot 119891 (119901) 119889119901 = int119863

119892 (119901) sdot 119891 (119901) 119889119901

(62)

where

119892 (119901) =AL (119901)119891 (119901)

(63)

Let region119863 be divided into119898 disjoint subregions119863119894 119894 =

1 119898 that is119863 = ⋃119898

119894=1119863119894and119863

119896cap119863119895= 0 119896 = 119895 where 0

is null setDefine

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901

119875119894= int119863119894

119891 (119901) 119889119901

119898

sum

119894=1

119875119894= 1

1198683= int119863

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 =

119898

sum

119894=1

1198683119894

(64)

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Probability and Statistics 13

Define

119892119894(1199011015840

) = 119892 (119901) 119901 isin 119863

119894

0 ow

1198683119894= int119863119894

119892 (119901) sdot 119891 (119901) 119889119901 = int119863119894

119875119894sdot 119892 (119901) sdot

119891 (119901)

119875119894

119889119901

= 119875119894int119863119894

119892 (1199011015840

) sdot119891 (119901)

119875119894

119889119901 = 119875119894119864 [119892119894(1199011015840

)]

(65)

where

int119863119894

119891 (119901)

119875119894

119889119901 = 1 (66)

Random variable 1199011015840 is distributed according to 119891(119901)119875119894

on119863119894

1198683119894can be estimated by

1198683119894=119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) 119894 = 1 119898 (67)

Therefore integral 1198683is estimated by

1198683=

119898

sum

119894=1

1198683119894=

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

) (68)

where 1199011015840119894is distributed according to 119891(119901)119875

119894on119863119894

997904rArr 119860 = minus1198683

119860 = minus

119898

sum

119894=1

119875119894

119873119894

119873119894

sum

119896119894=1

119892 (1199011015840

119896119894

)

Var (1198683) =

119898

sum

119894=1

1198752

119894

119873119894

sdot Var [119892 (1199011015840119894)]

=

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(69)

where

1205902

119894= Var[119892 (1199011015840

119894)] =

1

119875119894

int119863119894

1198922

(119901) sdot 119891 (119901) 119889119901 minus1198682

3119894

1198752

119894

Var (119860) = Var (minus1198683) = Var (119868

3) =

119898

sum

119894=1

1198752

1198941205902

119894

119873119894

(70)

In particular if 119875119894= 1119898 and 119873

119894= 119873119898 we get the

systematic sampling estimate of Absolute Lorenz index as

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (1199011015840

119870119894

) (71)

Algorithm for computing systematic random sampling esti-mate is given as follows

(i) Divide the range [0 1] of the cumulative distributioninto119898 intervals each of width 1119898

(ii) Generate 119880119896119894 119896119894= 1 119873119898 119894 = 1 119898 from

119880(0 1)

(iii) Compute 119884119896119894= (119894 minus 1 + 119880

119896119894)119898 119896

119894= 1 119873119898 119894 =

1 119898

(iv) Compute 119901119896119894= 119865minus1

(119884119896119894)

(v) Compute

119860 = minus1

119873

119898

sum

119894=1

119873119894

sum

119896119894=1

119892 (119901119896119894) (72)

Systematic random sampling estimates of Gini Bonferroniand Absolute Lorenz indices usingMonte Carlo approach aregiven in Table 3

3 Simulation Study

In this section we illustrate the performance of the estimatorsfor the three inequality indices based on the previouslymentioned sampling procedures 10000 random samples eachof size 6 8 10 12 15 20 25 30 60 80 100 120 300 400500 600 800 1000 1200 1500 2000 2500 and 3000 weregenerated to compare the ranked set and systematic samplingestimates with simple random sampling estimates of the Giniindex from exponential Pareto (120572 = 3) and Power (120572 = 3)Bonferroni index from exponential Pareto (120572 = 3) Power(120572 = 3) and Absolute Lorenz index from exponential (3)Pareto (120572 = 3) and rectangular (120579 = 3) distribution Thesimulated ranked set samples with set size 119898 = 3 4 5 6

and number of repetitions 119903 = 2 5 20 200 and 500 werechosenThemean squared errors (MSE) and efficiencies werecomputed for the three procedures Simulation results areshown in Tables 7 8 9 10 11 12 13 14 and 15

The expressions for Gini index Bonferroni and absoluteLorenz index are given in Tables 4 5 and 6 The relationbetween Pareto Power and Rectangular distribution can benoted from the following remark

Remark

(1) If random variable 119883 sim Pow(120572 120573) that is 119891(119909) =

120572120573120572

119909120572minus1

0 le 119909 le 120573 then 119884 = 1119883 sim Par(120572 1120573)that is 119891(119910) = (120572120573)(120573119910)120572+1 1120573 le 119910 le infin

(2) The Power function distribution Pow(120572 120573) withparameter 120572 = 1 and 120573 results in rectangulardistribution defined on the interval [0 1120573]

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 ISRN Probability and Statistics

For simple random sampling MSEs are defined as

MSE (SRS) =sum10000

119894=1(119866SRS minus 119866)

2

119873

MSE (SRS) =sum10000

119894=1(119861SRS minus 119861)

2

119873

MSE (SRS) =sum10000

119894=1(119860SRS minus 119860)

2

119873

(73)

where119866SRS 119861SRS and119860SRS denote the simple random sampleestimates of Gini Bonferroni and Absolute Lorenz indices(Table 1)

Similarly for ranked set and systematic sampling theMSEs can be defined where 119866RSS 119861RSS and 119860RSS denotethe ranked set sample estimates and 119866SYS 119861SYS and 119860SYSdenote the systematic random sample (SYS) estimates ofGiniBonferroni and Absolute Lorenz indices (Tables 2 and 3)

Gini index (119866) Bonferroni index (119861) and AbsoluteLorenz index (119860) for given distributions is given in the Tables4ndash6

Efficiencies are defined as

119864 (1) =MSE (SRS)MSE (RSS)

(74)

119864(1) gt 1 implies that Ranked set sample estimates are betterthan Simple random sample estimates

Similarly

119864 (2) =MSE (SRS)MSE (SYS)

(75)

119864(2) gt 1 implies that Systematic random sample (SYS)estimates are better than Simple random sample estimates

119864 (3) =MSE (RSS)MSE (SYS)

(76)

119864(3) gt 1 implies that Systematic random sample (SYS)estimates are better than Ranked set sample estimates

The results have been presented in Tables 7ndash15 and itcan be seen that the simple random estimates have muchless efficiency as compared to their ranked set or systematicsample estimates Systematic sample estimates are moreefficient than the ranked set sample estimates The sametrend is prevalent through all the distributions irrespectiveof the inequality index being used We also observe thatrelative efficiency increases as119898 increases irrespective of thesampling technique used

4 Conclusion

Ranked set sample estimates of inequality do improve theefficiency of all the inequality indices but compared tosystematic sampling the ranked set sample estimates are lessefficient For example in Table 7 while looking at the relativeefficiencies for Gini index using different sample estimates we

see that the efficiency of RSS is almost two to three timesmorethan SRS (ref Col119864(1)) while efficiency of SYS is almost fourto eight timesmore than RSS (ref Col119864(3))The efficiency ofSYS versus SRS is noticeably high as is evident from col 119864(2)The relative efficiency trend as given in the last three columnscontinues both for small as well as large sample sizes Onemay conclude that SRS estimates though commonly used andsimple to compute show less efficiency than their nonsimplerandom counterparts One should use non-simple randomestimate of inequality indices when the underlying data orpractical situation so demands

References

[1] M O Lorenz ldquoMethods of measuring the concentration ofwealthrdquo Publication of the American Statistical Association vol9 pp 209ndash219 1905

[2] C Gini ldquoMeasurement of inequality of incomesrdquoThe EconomicJournal vol 31 pp 124ndash126 1921

[3] A F Shorrocks ldquoRanking income distributionsrdquo Economicavol 50 pp 3ndash17 1983

[4] P Moyes ldquoA new concept of Lorenz dominationrdquo EconomicsLetters vol 23 no 2 pp 203ndash207 1987

[5] G M Giorgi Concentration Index Bonferroni Encyclopedia ofStatistical Sciences vol 2 John Wiley amp Sons New York NYUSA 1998

[6] S Arora K Jain and S Pundir ldquoOn cumulated mean incomecurverdquo Model Assisted Statistics and Applications vol 1 no 2pp 107ndash114 2006

[7] B Zheng ldquoTesting Lorenz curves with non-simple randomsamplesrdquo Econometrica vol 70 no 3 pp 1235ndash1243 2002

[8] G A Mcintyre ldquoA method for unbiased selective samplingusing ranked setsrdquo Australian Journal of Agricultural Researchvol 2 pp 385ndash390 1952

[9] M M Al-Talib and A D Al-Nasser ldquoEstimation of Gini-indexfrom continuous distribution based on ranked set samplingrdquoElectronic Journal of Applied Statistical Analysis vol 1 pp 33ndash41 2008

[10] R Rubinstein Simulation and the Monte Carlo Method JohnWiley amp Sons New York NY USA 1981

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of