Research ArticleEfficiency of Ratio Product and Regression Estimators underMaximum and Minimum Values Using Two Auxiliary Variables

Abdullah Y Al-Hossain1 and Mursala Khan2

1 Department of Mathematics Faculty of Science Jazan University Jazan 2097 Saudi Arabia2Department of Mathematics COMSATS Institute of Information Technology Abbottabad Pakistan

Correspondence should be addressed to Mursala Khan mursalakhanyahoocom

Received 4 December 2013 Accepted 9 February 2014 Published 13 April 2014

Academic Editor Bernard J Geurts

Copyright copy 2014 A Y Al-Hossain and M Khan This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

To obtain the best estimates of the unknown population parameters have been the key theme of the statisticians In the present paperwe have suggested some estimators which estimate the population parameters efficiently In short we propose a ratio product andregression estimators using two auxiliary variables when there are somemaximum andminimum values of the study and auxiliaryvariables respectively The properties of the proposed strategies in terms of mean square errors (variances) are derived up to firstorder of approximation Also the performance of the proposed estimators have shown theoretically and these theoretical conditionsare verified numerically by taking four real data sets under which the proposed class of estimators performed better than the otherprevious works

1 Introduction

In the literature of survey sampling the use of ancillaryinformation provided by auxiliary variables was discussed byvarious statisticians in order to improve the efficiency of theirconstructed estimators or to obtain improved estimators forestimating some most common population parameters suchas population mean population total population varianceand population coefficient of variation In such a situationratio product and regression estimators provide better esti-mates of the population parameters The work of Neyman [1]is considered as the early works where auxiliary informationhas been used After that a lot of work has been donefor estimating finite population mean and other populationparameters using auxiliary information and for improvingtheir efficiency For a more related work one can go throughDas and Tripathi [2 3] Upadhyaya and Singh [4] Singh[5] and so forth Sisodia and Dwivedi [6] have proposedratio estimator using coefficient of variation of an auxiliaryvariable Kadilar and Cingi [7] have suggested an estimatorfor population mean using two auxiliary variables Khan andShabbir [8] have introduced the idea of ratio type estimatoror the estimation of population variance using quartiles of

an auxiliary variable Mouatasim and Al-Hossain [9] havestudied reduced gradient method for minimax estimationof a bounded poisson mean in which concept of auxiliaryvariables can be easily placed and study Further Al-Hossain[10] has studied inference on compound Rayleigh parameterswith progressively type II censored samples wherein censoredsamples can be chosen as to auxiliary variables RecentlyKhan and Shabbir [11] suggested different estimators of finitepopulation mean using maximum and minimum values

Let us consider a finite population of size 119873 of differentunits 119880 = 119880

1 1198802 1198803 119880

119873 Let 119910 119909

1 and 119909

2be the

study and the auxiliary variables with corresponding values119910119894 1199091119894 and 119909

2119894 respectively for the 119894th unit 119894 = 1 2 3 119873

defined on a finite population 119880 Let 119884 = (1119873)sum119873

119894=1119910119894

1198831

= (1119873)sum119873

119894=11199091119894 and 119883

2= (1119873)sum

119873

119894=11199092119894

be thepopulation means of the study as well as auxiliary variablesrespectively let 1198782

119910= (1119873 minus 1)sum

119873

119894=1(119910119894minus 119884)2 11987821199091

= (1119873 minus

1)sum119873

119894=1(1199091119894minus 1198831)2 and 119878

2

1199092

= (1119873 minus 1)sum119873

119894=1(1199092119894minus 1198832)2

be the corresponding population variances of the study aswell as auxiliary variables respectively let 119862

119910 1198621199091

and1198621199092

be the coefficient of variation of the study as well as

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 693782 6 pageshttpdxdoiorg1011552014693782

2 Journal of Applied Mathematics

auxiliary variables respectively and let 1205881199101199091

1205881199101199092

and 12058811990911199092

be the population correlation coefficient among 119910 1199091 1199092and

between 1199091and 119909

2 respectively

In order to estimate the unknown population mean wetake a random sample of size 119899 units from the finite popula-tion119880 by using simple random sample without replacementLet 119910 119909

1 and 119909

2be the study and the auxiliary variables

with corresponding values 119910119894 1199091119894 and 119909

2119894 respectively for

the 119894th unit 119894 = 1 2 3 119899 in the sample Let 119910 =

(1119899)sum119899

119894=1119910119894 1199091= (1119899)sum

119899

119894=11199091119894 and 119909

2= (1119899)sum

119899

119894=11199092119894be

the sample means of the study as well as auxiliary variablesrespectively and let 119878

2

119910= (1119899 minus 1)sum

119899

119894=1(119910119894minus 119910)2 11987821199091

=

(1119899 minus 1)sum119899

119894=1(1199091119894minus 1199091)2 and 119878

2

1199092

= (1119899 minus 1)sum119899

119894=1(1199092119894minus 1199092)2

be the corresponding sample variances of the study as well asauxiliary variables respectively Also let 119862

119910 1198621199091

and 1198621199092

bethe sample coefficient of variation of the study variable 119910 aswell as auxiliary variables 119909

1and 119909

2 respectively and let 119878

1199101199091

1198781199101199092

and 11987811990911199092

be the sample covariances between 119910 1199091 and

1199092and between 119909

1and 119909

2 respectively

The usual unbiased estimator to estimate the populationmean of the study variable is

119910 =sum119899

119894=1119910119894

119899 (1)

The variance of the estimator 119910 up to first order of approxi-mation is given as follows

var (119910) = 1205791198782

119910 (2)

where 120579 = 1119899 minus 1119873In many real data sets there exist some large (119910max) or

small values (119910min) and to estimate the unknown populationparameters without considering this information is verysensitive in case the result will be either overestimated orunderestimated In order to handle this situation Sarndal [12]suggested the following unbiased estimator for the estimationof finite population mean using maximum and minimumvalues

119910119878=

119910 + 119888 if sample contains 119910min but not 119910max119910 minus 119888 if sample contains119910max but not 119910min119910 for all other samples

(3)

where 119888 is a constant which is to be found for minimumvariance

The minimum variance of the estimator 119910119878up to first

order of approximation is given as

var(119910119878)min = var (119910) minus

120579(119910max minus 119910min)2

2 (119873 minus 1) (4)

where the optimum value of 119888opt is

119888opt =(119910max minus 119910min)

2119899 (5)

The ratio estimator for estimating the unknown popula-tion mean of the study variable using two auxiliary variablesis given by

1198841198772

= 11991011988311198832

11990911199092

(6)

The mean square error of the estimator 1198841198772

up to firstorder of approximation is given by

MSE (1198841198772

) = 120579 [1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus 211987721198781199101199092

minus 211987711198781199101199091

]

(7)

The product estimator for estimating the unknown popu-lationmean of the study variable using two auxiliary variablesis given by

1198841198752

= 11991011990911199092

11988311198832

(8)

The mean square error of the estimator 1198841198752

up to firstorder of approximation is given by

MSE (1198841198752) = 120579 [119878

2

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

+ 211987721198781199101199092

+ 211987711198781199101199091

]

(9)

When there are two auxiliary variables then the regres-sion estimator to estimate the finite population mean is givenby

1198841198971199032

= 119910 + 1198871(1198831minus 1199091) + 1198872(1198832minus 1199092) (10)

where 1198871

= 1198781199101199091

1198782

1199091

and 1198872

= 1198781199101199092

1198782

1199092

are the sampleregression coefficients between 119910 and 119909

1and between 119910 and

1199092 respectivelyThe variance of the estimator

1198841198971199032

up to first order ofapproximation is given as

MSE (1198841198971199032

) = 1205791198782

119910[1 minus 120588

2

1199101199091

minus 1205882

1199101199092

+ 21205881199101199091

1205881199101199092

12058811990911199092

] (11)

2 Proposed Estimators

On the lines of Sarndal [12] we propose a ratio productand regression estimators using two auxiliary variables whenthere are some maximum and minimum values of the studyvariables and the auxiliary variables respectively

Case 1 When the correlation between the study variableand the auxiliary variable is positive the selection of thelarger value of the auxiliary variable the larger the value ofstudy variable is to be expected and the smaller the valueof auxiliary variable the smaller the value of study variable is

Journal of Applied Mathematics 3

to be expected and using such type of information the ratioestimator using two auxiliary variables becomes

1198841198771198622

= 11991011986211

1198831

119909111986221

1198832

119909211986231

=

(119910 + 1198881)

1198831

(1199091+ 1198882)

1198832

(1199092+ 1198883)

if sample contains 119910min and (1199091min 1199092min)

(119910 minus 1198881)

1198831

(1199091minus 1198882)

1198832

(1199092minus 1198883)

if sample contains 119910max and (1199091max 1199092max)

1199101198831

1199091

1198832

1199092

for all other samples

(12)

119884119897119903(1198751)

= 11991011986211

+ 1198871(1198831minus 119909111986221

) + 1198872(1198832minus 119909211986231

) (13)

where (11991011986211

= 119910 + 1198881 119909111986221

= 1199091+ 1198882 119909211986231

= 1199092+ 1198883) if

the sample contains 119910min and (1199091min 1199092min) (119910119862

12

= 119910 minus 1198881

119909211986222

= 1199091minus 1198882 119909211986232

= 1199092minus 1198883) if the sample contains 119910max

and (1199091max 1199092max) And (119910

11986211

= 119910 119909111986221

= 1199091 119909211986231

= 1199092)

for all other samples

Case 2 Similarly when the correlation is negative the selec-tion of the larger value of the auxiliary variable the smallerthe value of study variable is to be expected and the smallerthe value of auxiliary variable the larger the value of studyvariable is to be expected and using such type of informationthe product estimator using two auxiliary variables becomes

1198841198751198622

= 11991011986212

119909111986222

1198831

119909211986232

1198832

=

(119910 + 1198881)(1199091minus 1198882)

1198831

(1199092minus 1198883)

1198832

if sample contains 119910min and (1199091max 1199092max)

(119910 minus 1198881)(1199091+ 1198882)

1198831

(1199092+ 1198883)

1198832

if sample contains 119910max and (1199091min 1199092min)

1199101199091

1198831

1199092

1198832

for all other samples

119884119897119903(1198752)

= 11991011986212

+ 1198871(1198831minus 119909111986222

) + 1198872(1198832minus 119909211986232

)

(14)

where (11991011986212

= 119910 + 1198881 119909111986222

= 1199091minus 1198882 119909211986232

= 1199092minus 1198883) if

the sample contains 119910min and (1199091max 1199092max) (119910119862

11

= 119910 minus 1198881

119909111986221

= 1199091+ 1198882 119909211986231

= 1199092+ 1198883) if the sample contains 119910max

and (1199091min 1199092min) and (119910

11986212

= 119910 119909111986222

= 1199091 119909211986232

= 1199092) for

all other samples Also 1198881 1198882 and 119888

3are unknown constants

whose value is to be determined for optimality conditions

To obtain the properties of the proposed estimators interms of bias and mean square error we define the followingrelative error terms and their expectations

1198900= (1199101198881

minus 119884)119884 1198901= (11990911198882

minus 1198831)1198831 and 119890

2= (11990921198883

minus

1198832)1198832 such that 119864(119890

0) = 119864(119890

1) = 119864(119890

2) = 0 Consider also

119864 (1198902

0) =

120579

1198842(1198782

119910minus

21198991198881

119873 minus 1(119910max minus 119910min minus 119899119888

1))

119864 (1198902

1) =

120579

1198832

1

(1198782

1199091

minus21198991198882

119873 minus 1(1199091max minus 119909

1min minus 1198991198882))

119864 (1198902

2) =

120579

1198832

2

(1198782

1199092

minus21198991198883

119873 minus 1(1199092max minus 119909

2min minus 1198991198883))

119864 (11989001198901) =

120579

1198841198831

(1198781199101199091

minus119899

119873 minus 1

times (1198882(119910max minus 119910min)

+ 1198881(1199091max minus 119909

1min) minus 211989911988811198882) )

119864 (11989001198902) =

120579

1198841198832

(1198781199101199092

minus119899

119873 minus 1

times (1198883(119910max minus 119910min)

+ 1198881(1199092max minus 119909

2min) minus 211989911988811198883) )

119864 (11989011198902) =

120579

11988311198832

(11987811990911199092

minus119899

119873 minus 1

times (1198883(1199091max minus 119909

1min)

+ 1198882(1199092max minus 119909

2min) minus 211989911988821198883) )

(15)

Rewriting (12) 1198841198771198622

in terms of 119890119894rsquos we have

1198841198771198622

= 119884 (1 + 1198900) (1 + 119890

1)minus1

(1 + 1198902)minus1

(16)

Expanding the right hand side of above equation and includ-ing terms up to second powers of 119890

119894rsquos that is up to first order

of approximation we have

1198841198771198622

minus 119884 = 119884 (1198900minus 1198901minus 1198902+ 1198902

2+ 1198902

1+ 11989011198902minus 11989001198902minus 11989001198901)

(17)

On squaring both sides of (17) and keeping 119890119894rsquos powers up

to first order of approximation we have

(1198841198771198622

minus 119884)

2

= 1198842

[1198902

0+ 1198902

1+ 1198902

2minus 211989001198901minus 211989001198902+ 211989011198902]

(18)

4 Journal of Applied Mathematics

Taking expectation on both sides of (18) we get meansquare error up to first order of approximation given as

MSE (1198841198771198622

)

= [120579 (1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus 211987721198781199101199092

minus 211987711198781199101199091

) minus2119899120579 (119888

1minus 11987721198883minus 11987711198882)

119873 minus 1

times (119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 119899 (1198881minus 11987721198883minus 11987711198882) ]

(19)

To find the minimum mean squared error of 1198841198771198622

wedifferentiate (19) with respect to 119888

1 1198882 and 119888

3 respectively

that is

120597MSE (1198841198771198622

)

1205971198881

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198882

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198883

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

(20)

On differentiating (19) with respect to 1198881 1198882 and 119888

3

respectively we get one equation with three unknowns andso unique solution is not possible so let

1198881=

(119910max minus 119910min)

2119899

1198882=

(1199091max minus 119909

1min)

2119899

1198883=

(1199092max minus 119909

2min)

2119899

(21)

On substituting the optimum value of 1198881 1198882 and 119888

3from

(21) in (19) we get the minimum mean square error of theproposed estimator given as

MSE(1198841198771198622

)min

= MSE (1198772

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

minus 1198771(1199091max minus 119909

1min) minus 1198772(1199092max minus 119909

2min)]2

(22)

where MSE(1198841198772

) = 120579[1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus

211987721198781199101199092

minus 211987711198781199101199091

]Similarly the mean square error of the product estimator

up to first order of approximation is given by

MSE(1198841198751198622

)min

= MSE (1198752) minus

120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1198771(1199091max minus 119909

1min) + 1198772(1199092max minus 119909

2min)]2

(23)

where MSE(1198841198752) = 120579[119878

2

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

+

211987721198781199101199092

+ 211987711198781199101199091

]Now the minimum variance of the regression estimator

in the case of positive correlation up to first order ofapproximation is given by

MSE(119884119897119903(1198751)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus 1205731(1199091max minus 119909

1min) minus 1205732(1199092max minus 119909

2min))2

(24)

where MSE(1198841198971199032

) = 1205791198782

119910[1 minus 120588

2

1199101199091

minus 1205882

1199101199092

+ 21205881199101199091

1205881199101199092

12058811990911199092

]Similarly for the case of negative correlation the mini-

mum variance of the regression estimator up to first order ofapproximation is given by

MSE(119884119897119903(1198752)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1205731(1199091max minus 119909

1min) + 1205732(1199092max minus 119909

2min)]2

(25)

Journal of Applied Mathematics 5

But when there is positive and negative correlation theregression estimator gives us better result and so for bothcases (positive and negative correlation)wewrite the varianceas

MSE(119884119897119903(119875)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min) minus

100381610038161003816100381612057321003816100381610038161003816 (1199092max minus 119909

2min))2

(26)

3 Comparison of Estimators

In this section we have compared the proposed estimatorswith the ratio product and regression estimators and someof their efficiency comparison condition has been carried outunder which the proposed estimators perform better

(i) By (7) and (22)

[MSE (1198841198772

) minusMSE(1198841198771198622

)min

] ge 0 (27)

if

[(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus1198772(1199092max minus 119909

2min)]2

ge 0

(28)

(ii) By (9) and (23)

[MSE (1198841198752) minusMSE(119884

1198751198622)min

] ge 0 (29)

if

[(119910max minus 119910min) + 1198771(1199091max minus 119909

1min)

+ 1198772(1199092max minus 119909

2min)]2

ge 0

(30)

(iii) By (11) and (26)

[MSE (1198841198971199032

) minusMSE(119884119897119903(119875)

)min

] ge 0 (31)

if

[(119910max minus 119910min) minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min)

minus10038161003816100381610038161205732

1003816100381610038161003816 (1199092max minus 1199092min)]

2

ge 0

(32)

From (i) (ii) and (iii) we have observed that the proposedestimators performed better than the other existing estima-tors because the conditions are in the form of a square andgreater than zero which is always true

4 Numerical Illustration

In this section we demonstrate the performance of thesuggested estimators over various other estimators throughfour real data sets The description and the necessary datastatistics of the populations are given as follows

Population 1 (source Agricultural Statistics (1999) [13]Wash-ington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1993

119873 = 69 119899 = 20 1198831

= 4954435 1198832

= 4591072119884 = 4514899 1198782

119910= 37199578 1198782

1199092

= 39881874 1198622119910

=

18249 11986221199091

= 20300 11986221199092

= 18921 1199091max = 38007

1199091min = 32 119910max = 30027 119910min = 23 119909

2max = 340601199092min = 35 1198782

1199091

= 49829270 1205881199101199091

= 09601 1205881199101199092

=

09564 12058811990911199092

= 09729 1198781199101199091

= 4133593285 and 1198781199101199092

=

3683802614

Population 2 (source Agricultural Statistics (1998) [14]Washington US)

119884 season average price per pound during 19961198831 season average price per pound during 1995

1198832 season average price per pound during 1994

119873 = 36 119899 = 12 1198831

= 01856 1198832

= 01708119884 = 02033 119909

1max = 0403 1199091min = 0071 119910max =

0452 119910min = 0101 1199092max = 0334 119909

2min = 00781198782

119910= 0006458 1198782

1199091

= 0005654 11987821199092

= 0004017 1198622119910=

01563 11986221199091

= 01641 11986221199092

= 013757 1205881199101199091

= 087751205881199101199092

= 08577 12058811990911199092

= 08788 1198781199101199091

= 00053 and1198781199101199092

= 00044

Population 3 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4954435 119883

2= 4230174 119884 =

4514899 1199091max = 38007 119909

1min = 32 1198782119910= 37199578

119910max = 30027 119910min = 23 11987821199091

= 49829270 11987821199092

=

31010599 1199092max = 38933 119909

2min = 5 1198622119910

= 182491198622

1199091

= 20300 11986221199092

= 17329 1198781199101199091

= 41335932851198781199101199092

= 3239525507 11987811990911199092

= 3764276185 1205881199101199091

=

09601 1205881199101199092

= 09538 and 12058811990911199092

= 09576

Population 4 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1993

6 Journal of Applied Mathematics

Table 1 MSE of the existing and the proposed estimators

Estimator Population 1 Population 2 Population 3 Population 4MSE(sdot) MSE(sdot) MSE(sdot) MSE(sdot)

1198841198772

1671738947 00004 1513620253 14406684981198841198752

1216496392 00029 1177239482 11494548321198841198971199032

1254819885 00003 1217975563 1231973764Proposed

1198841198771198622

1293565557 00003 95766257 91198903871198841198751198622

9654413760 00021 8830692238 8616039070119884119897119903119875

9717815401 000025 7538683966 7694770316

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4591072 119883

2= 4230174 119884 =

4514899 119910max = 30027 119910min = 23 1199091max = 34060

1199091min = 35 1198782

119910= 37199578 1198782

1199091

= 39881874 11987821199092

=

31010599 1198622119910

= 18249 1199092max = 28933 119909

2min = 51198622

1199091

= 1892111986221199092

= 17329 1198781199101199091

= 3683802614 1205881199101199091

=

09564 11987811990911199092

= 3387344188 1198781199101199092

= 32395255071205881199101199092

= 09538 and 12058811990911199092

= 09632

The mean squared error of the proposed and the existingestimators is shown in Table 1

5 Conclusion and Future Work

We have developed some ratio product and regressionestimators under maximum and minimum values using twoauxiliary variables The proposed estimators under certainefficiency conditions are shown to be more efficient than theratio product and regression estimators using two auxiliaryvariables The results are shown numerically in Table 1 wherewe observed that the performance of the proposed estimatorsis better than the usual ratio product and the regressionestimators using two auxiliary variablesWe can easily imple-ment the concept of auxiliary variables minimax or maximinestimation of a bounded Poisson (respectively some otherdistribution) mean and censors samples Thus the proposedestimators may be preferred over the existing estimators forthe use of practical applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J Neyman ldquoContributions to the theory of sampling humanpopulationsrdquo Journal of the American Statistical Association vol33 no 201 pp 101ndash116 1938

[2] A K Das and T P Tripathi ldquoSampling strategies for populationmean when the coefficient of variation of an auxiliary characteris knownrdquo Sankhya vol 42 pp 76ndash86 1980

[3] A K Das and T P Tripathi ldquoA class of sampling strategies forpopulation mean using information on mean and variance of

an auxiliary characterrdquo in Proceedings of the Indian StatisticalInstitute Golden Jubilee International Conference on StatisticsApplications and New Directions pp 174ndash181 Calcutta IndiaDecember 1981

[4] L N Upadhyaya and H P Singh ldquoUse of transformed auxiliaryvariable in estimating the finite population meanrdquo BiometricalJournal vol 41 no 5 pp 627ndash636 1999

[5] S Singh ldquoGolden and Silver Jubilee year-2003 of the linearregression estimatorsrdquo in Proceedings of the American StatisticalAssociation Survey Method Section pp 4382ndash4389 AmericanStatistical Association Toronto Canada 2004

[6] B V S Sisodia and V K Dwivedi ldquoA modified ratio estimatorusing coefficient of variation of auxiliary variablerdquo Journal of theIndian Society of Agricultural Statistics vol 33 no 1 pp 13ndash181981

[7] C Kadilar and H Cingi ldquoA new estimator using two auxiliaryvariablesrdquo Applied Mathematics and Computation vol 162 no2 pp 901ndash908 2005

[8] M Khan and J Shabbir ldquoA ratio type estimator for theestimation of population variance using quartiles of an auxiliaryvariablerdquo Journal of Statistics Applications and Probability vol 2no 3 pp 157ndash162 2013

[9] A EMouatasim andAAl-Hossain ldquoReduced gradientmethodfor minimax estimation of a bounded poissonmeanrdquo Journal ofStatistics Advances in Theory and Applications vol 2 no 2 pp185ndash199 2009

[10] Y Al-Hossain ldquoInferences on compound rayleigh parameterswith progressively type-II censored samplesrdquoWorld Academy ofScience Engineering and Technology vol 76 pp 885ndash892 2013

[11] M Khan and J Shabbir ldquoSome improved ratio product andregression estimators of finite population mean when usingminimum and maximum valuesrdquo The Scientific World Journalvol 2013 Article ID 431868 7 pages 2013

[12] C E Sarndal ldquoSample survey theory vs general statistical the-ory estimation of the populationmeanrdquo International StatisticalInstitute vol 40 pp 1ndash12 1972

[13] httpwwwnassusdagovPublicationsAg Statistics1999indexasp

[14] httpwwwnassusdagovPublicationsAg Statistics

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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

auxiliary variables respectively and let 1205881199101199091

1205881199101199092

and 12058811990911199092

be the population correlation coefficient among 119910 1199091 1199092and

between 1199091and 119909

2 respectively

In order to estimate the unknown population mean wetake a random sample of size 119899 units from the finite popula-tion119880 by using simple random sample without replacementLet 119910 119909

1 and 119909

2be the study and the auxiliary variables

with corresponding values 119910119894 1199091119894 and 119909

2119894 respectively for

the 119894th unit 119894 = 1 2 3 119899 in the sample Let 119910 =

(1119899)sum119899

119894=1119910119894 1199091= (1119899)sum

119899

119894=11199091119894 and 119909

2= (1119899)sum

119899

119894=11199092119894be

the sample means of the study as well as auxiliary variablesrespectively and let 119878

2

119910= (1119899 minus 1)sum

119899

119894=1(119910119894minus 119910)2 11987821199091

=

(1119899 minus 1)sum119899

119894=1(1199091119894minus 1199091)2 and 119878

2

1199092

= (1119899 minus 1)sum119899

119894=1(1199092119894minus 1199092)2

be the corresponding sample variances of the study as well asauxiliary variables respectively Also let 119862

119910 1198621199091

and 1198621199092

bethe sample coefficient of variation of the study variable 119910 aswell as auxiliary variables 119909

1and 119909

2 respectively and let 119878

1199101199091

1198781199101199092

and 11987811990911199092

be the sample covariances between 119910 1199091 and

1199092and between 119909

1and 119909

2 respectively

The usual unbiased estimator to estimate the populationmean of the study variable is

119910 =sum119899

119894=1119910119894

119899 (1)

The variance of the estimator 119910 up to first order of approxi-mation is given as follows

var (119910) = 1205791198782

119910 (2)

where 120579 = 1119899 minus 1119873In many real data sets there exist some large (119910max) or

small values (119910min) and to estimate the unknown populationparameters without considering this information is verysensitive in case the result will be either overestimated orunderestimated In order to handle this situation Sarndal [12]suggested the following unbiased estimator for the estimationof finite population mean using maximum and minimumvalues

119910119878=

119910 + 119888 if sample contains 119910min but not 119910max119910 minus 119888 if sample contains119910max but not 119910min119910 for all other samples

(3)

where 119888 is a constant which is to be found for minimumvariance

The minimum variance of the estimator 119910119878up to first

order of approximation is given as

var(119910119878)min = var (119910) minus

120579(119910max minus 119910min)2

2 (119873 minus 1) (4)

where the optimum value of 119888opt is

119888opt =(119910max minus 119910min)

2119899 (5)

The ratio estimator for estimating the unknown popula-tion mean of the study variable using two auxiliary variablesis given by

1198841198772

= 11991011988311198832

11990911199092

(6)

The mean square error of the estimator 1198841198772

up to firstorder of approximation is given by

MSE (1198841198772

) = 120579 [1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus 211987721198781199101199092

minus 211987711198781199101199091

]

(7)

The product estimator for estimating the unknown popu-lationmean of the study variable using two auxiliary variablesis given by

1198841198752

= 11991011990911199092

11988311198832

(8)

The mean square error of the estimator 1198841198752

up to firstorder of approximation is given by

MSE (1198841198752) = 120579 [119878

2

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

+ 211987721198781199101199092

+ 211987711198781199101199091

]

(9)

When there are two auxiliary variables then the regres-sion estimator to estimate the finite population mean is givenby

1198841198971199032

= 119910 + 1198871(1198831minus 1199091) + 1198872(1198832minus 1199092) (10)

where 1198871

= 1198781199101199091

1198782

1199091

and 1198872

= 1198781199101199092

1198782

1199092

are the sampleregression coefficients between 119910 and 119909

1and between 119910 and

1199092 respectivelyThe variance of the estimator

1198841198971199032

up to first order ofapproximation is given as

MSE (1198841198971199032

) = 1205791198782

119910[1 minus 120588

2

1199101199091

minus 1205882

1199101199092

+ 21205881199101199091

1205881199101199092

12058811990911199092

] (11)

2 Proposed Estimators

On the lines of Sarndal [12] we propose a ratio productand regression estimators using two auxiliary variables whenthere are some maximum and minimum values of the studyvariables and the auxiliary variables respectively

Case 1 When the correlation between the study variableand the auxiliary variable is positive the selection of thelarger value of the auxiliary variable the larger the value ofstudy variable is to be expected and the smaller the valueof auxiliary variable the smaller the value of study variable is

Journal of Applied Mathematics 3

to be expected and using such type of information the ratioestimator using two auxiliary variables becomes

1198841198771198622

= 11991011986211

1198831

119909111986221

1198832

119909211986231

=

(119910 + 1198881)

1198831

(1199091+ 1198882)

1198832

(1199092+ 1198883)

if sample contains 119910min and (1199091min 1199092min)

(119910 minus 1198881)

1198831

(1199091minus 1198882)

1198832

(1199092minus 1198883)

if sample contains 119910max and (1199091max 1199092max)

1199101198831

1199091

1198832

1199092

for all other samples

(12)

119884119897119903(1198751)

= 11991011986211

+ 1198871(1198831minus 119909111986221

) + 1198872(1198832minus 119909211986231

) (13)

where (11991011986211

= 119910 + 1198881 119909111986221

= 1199091+ 1198882 119909211986231

= 1199092+ 1198883) if

the sample contains 119910min and (1199091min 1199092min) (119910119862

12

= 119910 minus 1198881

119909211986222

= 1199091minus 1198882 119909211986232

= 1199092minus 1198883) if the sample contains 119910max

and (1199091max 1199092max) And (119910

11986211

= 119910 119909111986221

= 1199091 119909211986231

= 1199092)

for all other samples

Case 2 Similarly when the correlation is negative the selec-tion of the larger value of the auxiliary variable the smallerthe value of study variable is to be expected and the smallerthe value of auxiliary variable the larger the value of studyvariable is to be expected and using such type of informationthe product estimator using two auxiliary variables becomes

1198841198751198622

= 11991011986212

119909111986222

1198831

119909211986232

1198832

=

(119910 + 1198881)(1199091minus 1198882)

1198831

(1199092minus 1198883)

1198832

if sample contains 119910min and (1199091max 1199092max)

(119910 minus 1198881)(1199091+ 1198882)

1198831

(1199092+ 1198883)

1198832

if sample contains 119910max and (1199091min 1199092min)

1199101199091

1198831

1199092

1198832

for all other samples

119884119897119903(1198752)

= 11991011986212

+ 1198871(1198831minus 119909111986222

) + 1198872(1198832minus 119909211986232

)

(14)

where (11991011986212

= 119910 + 1198881 119909111986222

= 1199091minus 1198882 119909211986232

= 1199092minus 1198883) if

the sample contains 119910min and (1199091max 1199092max) (119910119862

11

= 119910 minus 1198881

119909111986221

= 1199091+ 1198882 119909211986231

= 1199092+ 1198883) if the sample contains 119910max

and (1199091min 1199092min) and (119910

11986212

= 119910 119909111986222

= 1199091 119909211986232

= 1199092) for

all other samples Also 1198881 1198882 and 119888

3are unknown constants

whose value is to be determined for optimality conditions

To obtain the properties of the proposed estimators interms of bias and mean square error we define the followingrelative error terms and their expectations

1198900= (1199101198881

minus 119884)119884 1198901= (11990911198882

minus 1198831)1198831 and 119890

2= (11990921198883

minus

1198832)1198832 such that 119864(119890

0) = 119864(119890

1) = 119864(119890

2) = 0 Consider also

119864 (1198902

0) =

120579

1198842(1198782

119910minus

21198991198881

119873 minus 1(119910max minus 119910min minus 119899119888

1))

119864 (1198902

1) =

120579

1198832

1

(1198782

1199091

minus21198991198882

119873 minus 1(1199091max minus 119909

1min minus 1198991198882))

119864 (1198902

2) =

120579

1198832

2

(1198782

1199092

minus21198991198883

119873 minus 1(1199092max minus 119909

2min minus 1198991198883))

119864 (11989001198901) =

120579

1198841198831

(1198781199101199091

minus119899

119873 minus 1

times (1198882(119910max minus 119910min)

+ 1198881(1199091max minus 119909

1min) minus 211989911988811198882) )

119864 (11989001198902) =

120579

1198841198832

(1198781199101199092

minus119899

119873 minus 1

times (1198883(119910max minus 119910min)

+ 1198881(1199092max minus 119909

2min) minus 211989911988811198883) )

119864 (11989011198902) =

120579

11988311198832

(11987811990911199092

minus119899

119873 minus 1

times (1198883(1199091max minus 119909

1min)

+ 1198882(1199092max minus 119909

2min) minus 211989911988821198883) )

(15)

Rewriting (12) 1198841198771198622

in terms of 119890119894rsquos we have

1198841198771198622

= 119884 (1 + 1198900) (1 + 119890

1)minus1

(1 + 1198902)minus1

(16)

Expanding the right hand side of above equation and includ-ing terms up to second powers of 119890

119894rsquos that is up to first order

of approximation we have

1198841198771198622

minus 119884 = 119884 (1198900minus 1198901minus 1198902+ 1198902

2+ 1198902

1+ 11989011198902minus 11989001198902minus 11989001198901)

(17)

On squaring both sides of (17) and keeping 119890119894rsquos powers up

to first order of approximation we have

(1198841198771198622

minus 119884)

2

= 1198842

[1198902

0+ 1198902

1+ 1198902

2minus 211989001198901minus 211989001198902+ 211989011198902]

(18)

4 Journal of Applied Mathematics

Taking expectation on both sides of (18) we get meansquare error up to first order of approximation given as

MSE (1198841198771198622

)

= [120579 (1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus 211987721198781199101199092

minus 211987711198781199101199091

) minus2119899120579 (119888

1minus 11987721198883minus 11987711198882)

119873 minus 1

times (119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 119899 (1198881minus 11987721198883minus 11987711198882) ]

(19)

To find the minimum mean squared error of 1198841198771198622

wedifferentiate (19) with respect to 119888

1 1198882 and 119888

3 respectively

that is

120597MSE (1198841198771198622

)

1205971198881

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198882

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198883

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

(20)

On differentiating (19) with respect to 1198881 1198882 and 119888

3

respectively we get one equation with three unknowns andso unique solution is not possible so let

1198881=

(119910max minus 119910min)

2119899

1198882=

(1199091max minus 119909

1min)

2119899

1198883=

(1199092max minus 119909

2min)

2119899

(21)

On substituting the optimum value of 1198881 1198882 and 119888

3from

(21) in (19) we get the minimum mean square error of theproposed estimator given as

MSE(1198841198771198622

)min

= MSE (1198772

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

minus 1198771(1199091max minus 119909

1min) minus 1198772(1199092max minus 119909

2min)]2

(22)

where MSE(1198841198772

) = 120579[1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus

211987721198781199101199092

minus 211987711198781199101199091

]Similarly the mean square error of the product estimator

up to first order of approximation is given by

MSE(1198841198751198622

)min

= MSE (1198752) minus

120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1198771(1199091max minus 119909

1min) + 1198772(1199092max minus 119909

2min)]2

(23)

where MSE(1198841198752) = 120579[119878

2

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

+

211987721198781199101199092

+ 211987711198781199101199091

]Now the minimum variance of the regression estimator

in the case of positive correlation up to first order ofapproximation is given by

MSE(119884119897119903(1198751)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus 1205731(1199091max minus 119909

1min) minus 1205732(1199092max minus 119909

2min))2

(24)

where MSE(1198841198971199032

) = 1205791198782

119910[1 minus 120588

2

1199101199091

minus 1205882

1199101199092

+ 21205881199101199091

1205881199101199092

12058811990911199092

]Similarly for the case of negative correlation the mini-

mum variance of the regression estimator up to first order ofapproximation is given by

MSE(119884119897119903(1198752)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1205731(1199091max minus 119909

1min) + 1205732(1199092max minus 119909

2min)]2

(25)

Journal of Applied Mathematics 5

But when there is positive and negative correlation theregression estimator gives us better result and so for bothcases (positive and negative correlation)wewrite the varianceas

MSE(119884119897119903(119875)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min) minus

100381610038161003816100381612057321003816100381610038161003816 (1199092max minus 119909

2min))2

(26)

3 Comparison of Estimators

In this section we have compared the proposed estimatorswith the ratio product and regression estimators and someof their efficiency comparison condition has been carried outunder which the proposed estimators perform better

(i) By (7) and (22)

[MSE (1198841198772

) minusMSE(1198841198771198622

)min

] ge 0 (27)

if

[(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus1198772(1199092max minus 119909

2min)]2

ge 0

(28)

(ii) By (9) and (23)

[MSE (1198841198752) minusMSE(119884

1198751198622)min

] ge 0 (29)

if

[(119910max minus 119910min) + 1198771(1199091max minus 119909

1min)

+ 1198772(1199092max minus 119909

2min)]2

ge 0

(30)

(iii) By (11) and (26)

[MSE (1198841198971199032

) minusMSE(119884119897119903(119875)

)min

] ge 0 (31)

if

[(119910max minus 119910min) minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min)

minus10038161003816100381610038161205732

1003816100381610038161003816 (1199092max minus 1199092min)]

2

ge 0

(32)

From (i) (ii) and (iii) we have observed that the proposedestimators performed better than the other existing estima-tors because the conditions are in the form of a square andgreater than zero which is always true

4 Numerical Illustration

In this section we demonstrate the performance of thesuggested estimators over various other estimators throughfour real data sets The description and the necessary datastatistics of the populations are given as follows

Population 1 (source Agricultural Statistics (1999) [13]Wash-ington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1993

119873 = 69 119899 = 20 1198831

= 4954435 1198832

= 4591072119884 = 4514899 1198782

119910= 37199578 1198782

1199092

= 39881874 1198622119910

=

18249 11986221199091

= 20300 11986221199092

= 18921 1199091max = 38007

1199091min = 32 119910max = 30027 119910min = 23 119909

2max = 340601199092min = 35 1198782

1199091

= 49829270 1205881199101199091

= 09601 1205881199101199092

=

09564 12058811990911199092

= 09729 1198781199101199091

= 4133593285 and 1198781199101199092

=

3683802614

Population 2 (source Agricultural Statistics (1998) [14]Washington US)

119884 season average price per pound during 19961198831 season average price per pound during 1995

1198832 season average price per pound during 1994

119873 = 36 119899 = 12 1198831

= 01856 1198832

= 01708119884 = 02033 119909

1max = 0403 1199091min = 0071 119910max =

0452 119910min = 0101 1199092max = 0334 119909

2min = 00781198782

119910= 0006458 1198782

1199091

= 0005654 11987821199092

= 0004017 1198622119910=

01563 11986221199091

= 01641 11986221199092

= 013757 1205881199101199091

= 087751205881199101199092

= 08577 12058811990911199092

= 08788 1198781199101199091

= 00053 and1198781199101199092

= 00044

Population 3 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4954435 119883

2= 4230174 119884 =

4514899 1199091max = 38007 119909

1min = 32 1198782119910= 37199578

119910max = 30027 119910min = 23 11987821199091

= 49829270 11987821199092

=

31010599 1199092max = 38933 119909

2min = 5 1198622119910

= 182491198622

1199091

= 20300 11986221199092

= 17329 1198781199101199091

= 41335932851198781199101199092

= 3239525507 11987811990911199092

= 3764276185 1205881199101199091

=

09601 1205881199101199092

= 09538 and 12058811990911199092

= 09576

Population 4 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1993

6 Journal of Applied Mathematics

Table 1 MSE of the existing and the proposed estimators

Estimator Population 1 Population 2 Population 3 Population 4MSE(sdot) MSE(sdot) MSE(sdot) MSE(sdot)

1198841198772

1671738947 00004 1513620253 14406684981198841198752

1216496392 00029 1177239482 11494548321198841198971199032

1254819885 00003 1217975563 1231973764Proposed

1198841198771198622

1293565557 00003 95766257 91198903871198841198751198622

9654413760 00021 8830692238 8616039070119884119897119903119875

9717815401 000025 7538683966 7694770316

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4591072 119883

2= 4230174 119884 =

4514899 119910max = 30027 119910min = 23 1199091max = 34060

1199091min = 35 1198782

119910= 37199578 1198782

1199091

= 39881874 11987821199092

=

31010599 1198622119910

= 18249 1199092max = 28933 119909

2min = 51198622

1199091

= 1892111986221199092

= 17329 1198781199101199091

= 3683802614 1205881199101199091

=

09564 11987811990911199092

= 3387344188 1198781199101199092

= 32395255071205881199101199092

= 09538 and 12058811990911199092

= 09632

The mean squared error of the proposed and the existingestimators is shown in Table 1

5 Conclusion and Future Work

We have developed some ratio product and regressionestimators under maximum and minimum values using twoauxiliary variables The proposed estimators under certainefficiency conditions are shown to be more efficient than theratio product and regression estimators using two auxiliaryvariables The results are shown numerically in Table 1 wherewe observed that the performance of the proposed estimatorsis better than the usual ratio product and the regressionestimators using two auxiliary variablesWe can easily imple-ment the concept of auxiliary variables minimax or maximinestimation of a bounded Poisson (respectively some otherdistribution) mean and censors samples Thus the proposedestimators may be preferred over the existing estimators forthe use of practical applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J Neyman ldquoContributions to the theory of sampling humanpopulationsrdquo Journal of the American Statistical Association vol33 no 201 pp 101ndash116 1938

[2] A K Das and T P Tripathi ldquoSampling strategies for populationmean when the coefficient of variation of an auxiliary characteris knownrdquo Sankhya vol 42 pp 76ndash86 1980

[3] A K Das and T P Tripathi ldquoA class of sampling strategies forpopulation mean using information on mean and variance of

an auxiliary characterrdquo in Proceedings of the Indian StatisticalInstitute Golden Jubilee International Conference on StatisticsApplications and New Directions pp 174ndash181 Calcutta IndiaDecember 1981

[4] L N Upadhyaya and H P Singh ldquoUse of transformed auxiliaryvariable in estimating the finite population meanrdquo BiometricalJournal vol 41 no 5 pp 627ndash636 1999

[5] S Singh ldquoGolden and Silver Jubilee year-2003 of the linearregression estimatorsrdquo in Proceedings of the American StatisticalAssociation Survey Method Section pp 4382ndash4389 AmericanStatistical Association Toronto Canada 2004

[6] B V S Sisodia and V K Dwivedi ldquoA modified ratio estimatorusing coefficient of variation of auxiliary variablerdquo Journal of theIndian Society of Agricultural Statistics vol 33 no 1 pp 13ndash181981

[7] C Kadilar and H Cingi ldquoA new estimator using two auxiliaryvariablesrdquo Applied Mathematics and Computation vol 162 no2 pp 901ndash908 2005

[8] M Khan and J Shabbir ldquoA ratio type estimator for theestimation of population variance using quartiles of an auxiliaryvariablerdquo Journal of Statistics Applications and Probability vol 2no 3 pp 157ndash162 2013

[9] A EMouatasim andAAl-Hossain ldquoReduced gradientmethodfor minimax estimation of a bounded poissonmeanrdquo Journal ofStatistics Advances in Theory and Applications vol 2 no 2 pp185ndash199 2009

[10] Y Al-Hossain ldquoInferences on compound rayleigh parameterswith progressively type-II censored samplesrdquoWorld Academy ofScience Engineering and Technology vol 76 pp 885ndash892 2013

[11] M Khan and J Shabbir ldquoSome improved ratio product andregression estimators of finite population mean when usingminimum and maximum valuesrdquo The Scientific World Journalvol 2013 Article ID 431868 7 pages 2013

[12] C E Sarndal ldquoSample survey theory vs general statistical the-ory estimation of the populationmeanrdquo International StatisticalInstitute vol 40 pp 1ndash12 1972

[13] httpwwwnassusdagovPublicationsAg Statistics1999indexasp

[14] httpwwwnassusdagovPublicationsAg Statistics

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

Journal of Applied Mathematics 3

to be expected and using such type of information the ratioestimator using two auxiliary variables becomes

1198841198771198622

= 11991011986211

1198831

119909111986221

1198832

119909211986231

=

(119910 + 1198881)

1198831

(1199091+ 1198882)

1198832

(1199092+ 1198883)

if sample contains 119910min and (1199091min 1199092min)

(119910 minus 1198881)

1198831

(1199091minus 1198882)

1198832

(1199092minus 1198883)

if sample contains 119910max and (1199091max 1199092max)

1199101198831

1199091

1198832

1199092

for all other samples

(12)

119884119897119903(1198751)

= 11991011986211

+ 1198871(1198831minus 119909111986221

) + 1198872(1198832minus 119909211986231

) (13)

where (11991011986211

= 119910 + 1198881 119909111986221

= 1199091+ 1198882 119909211986231

= 1199092+ 1198883) if

the sample contains 119910min and (1199091min 1199092min) (119910119862

12

= 119910 minus 1198881

119909211986222

= 1199091minus 1198882 119909211986232

= 1199092minus 1198883) if the sample contains 119910max

and (1199091max 1199092max) And (119910

11986211

= 119910 119909111986221

= 1199091 119909211986231

= 1199092)

for all other samples

Case 2 Similarly when the correlation is negative the selec-tion of the larger value of the auxiliary variable the smallerthe value of study variable is to be expected and the smallerthe value of auxiliary variable the larger the value of studyvariable is to be expected and using such type of informationthe product estimator using two auxiliary variables becomes

1198841198751198622

= 11991011986212

119909111986222

1198831

119909211986232

1198832

=

(119910 + 1198881)(1199091minus 1198882)

1198831

(1199092minus 1198883)

1198832

if sample contains 119910min and (1199091max 1199092max)

(119910 minus 1198881)(1199091+ 1198882)

1198831

(1199092+ 1198883)

1198832

if sample contains 119910max and (1199091min 1199092min)

1199101199091

1198831

1199092

1198832

for all other samples

119884119897119903(1198752)

= 11991011986212

+ 1198871(1198831minus 119909111986222

) + 1198872(1198832minus 119909211986232

)

(14)

where (11991011986212

= 119910 + 1198881 119909111986222

= 1199091minus 1198882 119909211986232

= 1199092minus 1198883) if

the sample contains 119910min and (1199091max 1199092max) (119910119862

11

= 119910 minus 1198881

119909111986221

= 1199091+ 1198882 119909211986231

= 1199092+ 1198883) if the sample contains 119910max

and (1199091min 1199092min) and (119910

11986212

= 119910 119909111986222

= 1199091 119909211986232

= 1199092) for

all other samples Also 1198881 1198882 and 119888

3are unknown constants

whose value is to be determined for optimality conditions

To obtain the properties of the proposed estimators interms of bias and mean square error we define the followingrelative error terms and their expectations

1198900= (1199101198881

minus 119884)119884 1198901= (11990911198882

minus 1198831)1198831 and 119890

2= (11990921198883

minus

1198832)1198832 such that 119864(119890

0) = 119864(119890

1) = 119864(119890

2) = 0 Consider also

119864 (1198902

0) =

120579

1198842(1198782

119910minus

21198991198881

119873 minus 1(119910max minus 119910min minus 119899119888

1))

119864 (1198902

1) =

120579

1198832

1

(1198782

1199091

minus21198991198882

119873 minus 1(1199091max minus 119909

1min minus 1198991198882))

119864 (1198902

2) =

120579

1198832

2

(1198782

1199092

minus21198991198883

119873 minus 1(1199092max minus 119909

2min minus 1198991198883))

119864 (11989001198901) =

120579

1198841198831

(1198781199101199091

minus119899

119873 minus 1

times (1198882(119910max minus 119910min)

+ 1198881(1199091max minus 119909

1min) minus 211989911988811198882) )

119864 (11989001198902) =

120579

1198841198832

(1198781199101199092

minus119899

119873 minus 1

times (1198883(119910max minus 119910min)

+ 1198881(1199092max minus 119909

2min) minus 211989911988811198883) )

119864 (11989011198902) =

120579

11988311198832

(11987811990911199092

minus119899

119873 minus 1

times (1198883(1199091max minus 119909

1min)

+ 1198882(1199092max minus 119909

2min) minus 211989911988821198883) )

(15)

Rewriting (12) 1198841198771198622

in terms of 119890119894rsquos we have

1198841198771198622

= 119884 (1 + 1198900) (1 + 119890

1)minus1

(1 + 1198902)minus1

(16)

Expanding the right hand side of above equation and includ-ing terms up to second powers of 119890

119894rsquos that is up to first order

of approximation we have

1198841198771198622

minus 119884 = 119884 (1198900minus 1198901minus 1198902+ 1198902

2+ 1198902

1+ 11989011198902minus 11989001198902minus 11989001198901)

(17)

On squaring both sides of (17) and keeping 119890119894rsquos powers up

to first order of approximation we have

(1198841198771198622

minus 119884)

2

= 1198842

[1198902

0+ 1198902

1+ 1198902

2minus 211989001198901minus 211989001198902+ 211989011198902]

(18)

4 Journal of Applied Mathematics

Taking expectation on both sides of (18) we get meansquare error up to first order of approximation given as

MSE (1198841198771198622

)

= [120579 (1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus 211987721198781199101199092

minus 211987711198781199101199091

) minus2119899120579 (119888

1minus 11987721198883minus 11987711198882)

119873 minus 1

times (119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 119899 (1198881minus 11987721198883minus 11987711198882) ]

(19)

To find the minimum mean squared error of 1198841198771198622

wedifferentiate (19) with respect to 119888

1 1198882 and 119888

3 respectively

that is

120597MSE (1198841198771198622

)

1205971198881

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198882

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198883

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

(20)

On differentiating (19) with respect to 1198881 1198882 and 119888

3

respectively we get one equation with three unknowns andso unique solution is not possible so let

1198881=

(119910max minus 119910min)

2119899

1198882=

(1199091max minus 119909

1min)

2119899

1198883=

(1199092max minus 119909

2min)

2119899

(21)

On substituting the optimum value of 1198881 1198882 and 119888

3from

(21) in (19) we get the minimum mean square error of theproposed estimator given as

MSE(1198841198771198622

)min

= MSE (1198772

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

minus 1198771(1199091max minus 119909

1min) minus 1198772(1199092max minus 119909

2min)]2

(22)

where MSE(1198841198772

) = 120579[1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus

211987721198781199101199092

minus 211987711198781199101199091

]Similarly the mean square error of the product estimator

up to first order of approximation is given by

MSE(1198841198751198622

)min

= MSE (1198752) minus

120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1198771(1199091max minus 119909

1min) + 1198772(1199092max minus 119909

2min)]2

(23)

where MSE(1198841198752) = 120579[119878

2

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

+

211987721198781199101199092

+ 211987711198781199101199091

]Now the minimum variance of the regression estimator

in the case of positive correlation up to first order ofapproximation is given by

MSE(119884119897119903(1198751)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus 1205731(1199091max minus 119909

1min) minus 1205732(1199092max minus 119909

2min))2

(24)

where MSE(1198841198971199032

) = 1205791198782

119910[1 minus 120588

2

1199101199091

minus 1205882

1199101199092

+ 21205881199101199091

1205881199101199092

12058811990911199092

]Similarly for the case of negative correlation the mini-

mum variance of the regression estimator up to first order ofapproximation is given by

MSE(119884119897119903(1198752)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1205731(1199091max minus 119909

1min) + 1205732(1199092max minus 119909

2min)]2

(25)

Journal of Applied Mathematics 5

But when there is positive and negative correlation theregression estimator gives us better result and so for bothcases (positive and negative correlation)wewrite the varianceas

MSE(119884119897119903(119875)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min) minus

100381610038161003816100381612057321003816100381610038161003816 (1199092max minus 119909

2min))2

(26)

3 Comparison of Estimators

In this section we have compared the proposed estimatorswith the ratio product and regression estimators and someof their efficiency comparison condition has been carried outunder which the proposed estimators perform better

(i) By (7) and (22)

[MSE (1198841198772

) minusMSE(1198841198771198622

)min

] ge 0 (27)

if

[(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus1198772(1199092max minus 119909

2min)]2

ge 0

(28)

(ii) By (9) and (23)

[MSE (1198841198752) minusMSE(119884

1198751198622)min

] ge 0 (29)

if

[(119910max minus 119910min) + 1198771(1199091max minus 119909

1min)

+ 1198772(1199092max minus 119909

2min)]2

ge 0

(30)

(iii) By (11) and (26)

[MSE (1198841198971199032

) minusMSE(119884119897119903(119875)

)min

] ge 0 (31)

if

[(119910max minus 119910min) minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min)

minus10038161003816100381610038161205732

1003816100381610038161003816 (1199092max minus 1199092min)]

2

ge 0

(32)

From (i) (ii) and (iii) we have observed that the proposedestimators performed better than the other existing estima-tors because the conditions are in the form of a square andgreater than zero which is always true

4 Numerical Illustration

In this section we demonstrate the performance of thesuggested estimators over various other estimators throughfour real data sets The description and the necessary datastatistics of the populations are given as follows

Population 1 (source Agricultural Statistics (1999) [13]Wash-ington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1993

119873 = 69 119899 = 20 1198831

= 4954435 1198832

= 4591072119884 = 4514899 1198782

119910= 37199578 1198782

1199092

= 39881874 1198622119910

=

18249 11986221199091

= 20300 11986221199092

= 18921 1199091max = 38007

1199091min = 32 119910max = 30027 119910min = 23 119909

2max = 340601199092min = 35 1198782

1199091

= 49829270 1205881199101199091

= 09601 1205881199101199092

=

09564 12058811990911199092

= 09729 1198781199101199091

= 4133593285 and 1198781199101199092

=

3683802614

Population 2 (source Agricultural Statistics (1998) [14]Washington US)

119884 season average price per pound during 19961198831 season average price per pound during 1995

1198832 season average price per pound during 1994

119873 = 36 119899 = 12 1198831

= 01856 1198832

= 01708119884 = 02033 119909

1max = 0403 1199091min = 0071 119910max =

0452 119910min = 0101 1199092max = 0334 119909

2min = 00781198782

119910= 0006458 1198782

1199091

= 0005654 11987821199092

= 0004017 1198622119910=

01563 11986221199091

= 01641 11986221199092

= 013757 1205881199101199091

= 087751205881199101199092

= 08577 12058811990911199092

= 08788 1198781199101199091

= 00053 and1198781199101199092

= 00044

Population 3 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4954435 119883

2= 4230174 119884 =

4514899 1199091max = 38007 119909

1min = 32 1198782119910= 37199578

119910max = 30027 119910min = 23 11987821199091

= 49829270 11987821199092

=

31010599 1199092max = 38933 119909

2min = 5 1198622119910

= 182491198622

1199091

= 20300 11986221199092

= 17329 1198781199101199091

= 41335932851198781199101199092

= 3239525507 11987811990911199092

= 3764276185 1205881199101199091

=

09601 1205881199101199092

= 09538 and 12058811990911199092

= 09576

Population 4 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1993

6 Journal of Applied Mathematics

Table 1 MSE of the existing and the proposed estimators

Estimator Population 1 Population 2 Population 3 Population 4MSE(sdot) MSE(sdot) MSE(sdot) MSE(sdot)

1198841198772

1671738947 00004 1513620253 14406684981198841198752

1216496392 00029 1177239482 11494548321198841198971199032

1254819885 00003 1217975563 1231973764Proposed

1198841198771198622

1293565557 00003 95766257 91198903871198841198751198622

9654413760 00021 8830692238 8616039070119884119897119903119875

9717815401 000025 7538683966 7694770316

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4591072 119883

2= 4230174 119884 =

4514899 119910max = 30027 119910min = 23 1199091max = 34060

1199091min = 35 1198782

119910= 37199578 1198782

1199091

= 39881874 11987821199092

=

31010599 1198622119910

= 18249 1199092max = 28933 119909

2min = 51198622

1199091

= 1892111986221199092

= 17329 1198781199101199091

= 3683802614 1205881199101199091

=

09564 11987811990911199092

= 3387344188 1198781199101199092

= 32395255071205881199101199092

= 09538 and 12058811990911199092

= 09632

The mean squared error of the proposed and the existingestimators is shown in Table 1

5 Conclusion and Future Work

We have developed some ratio product and regressionestimators under maximum and minimum values using twoauxiliary variables The proposed estimators under certainefficiency conditions are shown to be more efficient than theratio product and regression estimators using two auxiliaryvariables The results are shown numerically in Table 1 wherewe observed that the performance of the proposed estimatorsis better than the usual ratio product and the regressionestimators using two auxiliary variablesWe can easily imple-ment the concept of auxiliary variables minimax or maximinestimation of a bounded Poisson (respectively some otherdistribution) mean and censors samples Thus the proposedestimators may be preferred over the existing estimators forthe use of practical applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J Neyman ldquoContributions to the theory of sampling humanpopulationsrdquo Journal of the American Statistical Association vol33 no 201 pp 101ndash116 1938

[2] A K Das and T P Tripathi ldquoSampling strategies for populationmean when the coefficient of variation of an auxiliary characteris knownrdquo Sankhya vol 42 pp 76ndash86 1980

[3] A K Das and T P Tripathi ldquoA class of sampling strategies forpopulation mean using information on mean and variance of

an auxiliary characterrdquo in Proceedings of the Indian StatisticalInstitute Golden Jubilee International Conference on StatisticsApplications and New Directions pp 174ndash181 Calcutta IndiaDecember 1981

[4] L N Upadhyaya and H P Singh ldquoUse of transformed auxiliaryvariable in estimating the finite population meanrdquo BiometricalJournal vol 41 no 5 pp 627ndash636 1999

[5] S Singh ldquoGolden and Silver Jubilee year-2003 of the linearregression estimatorsrdquo in Proceedings of the American StatisticalAssociation Survey Method Section pp 4382ndash4389 AmericanStatistical Association Toronto Canada 2004

[6] B V S Sisodia and V K Dwivedi ldquoA modified ratio estimatorusing coefficient of variation of auxiliary variablerdquo Journal of theIndian Society of Agricultural Statistics vol 33 no 1 pp 13ndash181981

[7] C Kadilar and H Cingi ldquoA new estimator using two auxiliaryvariablesrdquo Applied Mathematics and Computation vol 162 no2 pp 901ndash908 2005

[8] M Khan and J Shabbir ldquoA ratio type estimator for theestimation of population variance using quartiles of an auxiliaryvariablerdquo Journal of Statistics Applications and Probability vol 2no 3 pp 157ndash162 2013

[9] A EMouatasim andAAl-Hossain ldquoReduced gradientmethodfor minimax estimation of a bounded poissonmeanrdquo Journal ofStatistics Advances in Theory and Applications vol 2 no 2 pp185ndash199 2009

[10] Y Al-Hossain ldquoInferences on compound rayleigh parameterswith progressively type-II censored samplesrdquoWorld Academy ofScience Engineering and Technology vol 76 pp 885ndash892 2013

[11] M Khan and J Shabbir ldquoSome improved ratio product andregression estimators of finite population mean when usingminimum and maximum valuesrdquo The Scientific World Journalvol 2013 Article ID 431868 7 pages 2013

[12] C E Sarndal ldquoSample survey theory vs general statistical the-ory estimation of the populationmeanrdquo International StatisticalInstitute vol 40 pp 1ndash12 1972

[13] httpwwwnassusdagovPublicationsAg Statistics1999indexasp

[14] httpwwwnassusdagovPublicationsAg Statistics

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

4 Journal of Applied Mathematics

Taking expectation on both sides of (18) we get meansquare error up to first order of approximation given as

MSE (1198841198771198622

)

= [120579 (1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus 211987721198781199101199092

minus 211987711198781199101199091

) minus2119899120579 (119888

1minus 11987721198883minus 11987711198882)

119873 minus 1

times (119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 119899 (1198881minus 11987721198883minus 11987711198882) ]

(19)

To find the minimum mean squared error of 1198841198771198622

wedifferentiate (19) with respect to 119888

1 1198882 and 119888

3 respectively

that is

120597MSE (1198841198771198622

)

1205971198881

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198882

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

120597MSE (1198841198771198622

)

1205971198883

= 0 or

(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus 1198772(1199092max minus 119909

2min) minus 2119899 (1198881minus 11987711198882minus 11987721198883) = 0

(20)

On differentiating (19) with respect to 1198881 1198882 and 119888

3

respectively we get one equation with three unknowns andso unique solution is not possible so let

1198881=

(119910max minus 119910min)

2119899

1198882=

(1199091max minus 119909

1min)

2119899

1198883=

(1199092max minus 119909

2min)

2119899

(21)

On substituting the optimum value of 1198881 1198882 and 119888

3from

(21) in (19) we get the minimum mean square error of theproposed estimator given as

MSE(1198841198771198622

)min

= MSE (1198772

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

minus 1198771(1199091max minus 119909

1min) minus 1198772(1199092max minus 119909

2min)]2

(22)

where MSE(1198841198772

) = 120579[1198782

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

minus

211987721198781199101199092

minus 211987711198781199101199091

]Similarly the mean square error of the product estimator

up to first order of approximation is given by

MSE(1198841198751198622

)min

= MSE (1198752) minus

120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1198771(1199091max minus 119909

1min) + 1198772(1199092max minus 119909

2min)]2

(23)

where MSE(1198841198752) = 120579[119878

2

119910+ 1198772

11198782

1199091

+ 1198772

21198782

1199092

+ 21198771119877211987811990911199092

+

211987721198781199101199092

+ 211987711198781199101199091

]Now the minimum variance of the regression estimator

in the case of positive correlation up to first order ofapproximation is given by

MSE(119884119897119903(1198751)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus 1205731(1199091max minus 119909

1min) minus 1205732(1199092max minus 119909

2min))2

(24)

where MSE(1198841198971199032

) = 1205791198782

119910[1 minus 120588

2

1199101199091

minus 1205882

1199101199092

+ 21205881199101199091

1205881199101199092

12058811990911199092

]Similarly for the case of negative correlation the mini-

mum variance of the regression estimator up to first order ofapproximation is given by

MSE(119884119897119903(1198752)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times [(119910max minus 119910min)

+ 1205731(1199091max minus 119909

1min) + 1205732(1199092max minus 119909

2min)]2

(25)

Journal of Applied Mathematics 5

But when there is positive and negative correlation theregression estimator gives us better result and so for bothcases (positive and negative correlation)wewrite the varianceas

MSE(119884119897119903(119875)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min) minus

100381610038161003816100381612057321003816100381610038161003816 (1199092max minus 119909

2min))2

(26)

3 Comparison of Estimators

In this section we have compared the proposed estimatorswith the ratio product and regression estimators and someof their efficiency comparison condition has been carried outunder which the proposed estimators perform better

(i) By (7) and (22)

[MSE (1198841198772

) minusMSE(1198841198771198622

)min

] ge 0 (27)

if

[(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus1198772(1199092max minus 119909

2min)]2

ge 0

(28)

(ii) By (9) and (23)

[MSE (1198841198752) minusMSE(119884

1198751198622)min

] ge 0 (29)

if

[(119910max minus 119910min) + 1198771(1199091max minus 119909

1min)

+ 1198772(1199092max minus 119909

2min)]2

ge 0

(30)

(iii) By (11) and (26)

[MSE (1198841198971199032

) minusMSE(119884119897119903(119875)

)min

] ge 0 (31)

if

[(119910max minus 119910min) minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min)

minus10038161003816100381610038161205732

1003816100381610038161003816 (1199092max minus 1199092min)]

2

ge 0

(32)

From (i) (ii) and (iii) we have observed that the proposedestimators performed better than the other existing estima-tors because the conditions are in the form of a square andgreater than zero which is always true

4 Numerical Illustration

In this section we demonstrate the performance of thesuggested estimators over various other estimators throughfour real data sets The description and the necessary datastatistics of the populations are given as follows

Population 1 (source Agricultural Statistics (1999) [13]Wash-ington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1993

119873 = 69 119899 = 20 1198831

= 4954435 1198832

= 4591072119884 = 4514899 1198782

119910= 37199578 1198782

1199092

= 39881874 1198622119910

=

18249 11986221199091

= 20300 11986221199092

= 18921 1199091max = 38007

1199091min = 32 119910max = 30027 119910min = 23 119909

2max = 340601199092min = 35 1198782

1199091

= 49829270 1205881199101199091

= 09601 1205881199101199092

=

09564 12058811990911199092

= 09729 1198781199101199091

= 4133593285 and 1198781199101199092

=

3683802614

Population 2 (source Agricultural Statistics (1998) [14]Washington US)

119884 season average price per pound during 19961198831 season average price per pound during 1995

1198832 season average price per pound during 1994

119873 = 36 119899 = 12 1198831

= 01856 1198832

= 01708119884 = 02033 119909

1max = 0403 1199091min = 0071 119910max =

0452 119910min = 0101 1199092max = 0334 119909

2min = 00781198782

119910= 0006458 1198782

1199091

= 0005654 11987821199092

= 0004017 1198622119910=

01563 11986221199091

= 01641 11986221199092

= 013757 1205881199101199091

= 087751205881199101199092

= 08577 12058811990911199092

= 08788 1198781199101199091

= 00053 and1198781199101199092

= 00044

Population 3 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4954435 119883

2= 4230174 119884 =

4514899 1199091max = 38007 119909

1min = 32 1198782119910= 37199578

119910max = 30027 119910min = 23 11987821199091

= 49829270 11987821199092

=

31010599 1199092max = 38933 119909

2min = 5 1198622119910

= 182491198622

1199091

= 20300 11986221199092

= 17329 1198781199101199091

= 41335932851198781199101199092

= 3239525507 11987811990911199092

= 3764276185 1205881199101199091

=

09601 1205881199101199092

= 09538 and 12058811990911199092

= 09576

Population 4 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1993

6 Journal of Applied Mathematics

Table 1 MSE of the existing and the proposed estimators

Estimator Population 1 Population 2 Population 3 Population 4MSE(sdot) MSE(sdot) MSE(sdot) MSE(sdot)

1198841198772

1671738947 00004 1513620253 14406684981198841198752

1216496392 00029 1177239482 11494548321198841198971199032

1254819885 00003 1217975563 1231973764Proposed

1198841198771198622

1293565557 00003 95766257 91198903871198841198751198622

9654413760 00021 8830692238 8616039070119884119897119903119875

9717815401 000025 7538683966 7694770316

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4591072 119883

2= 4230174 119884 =

4514899 119910max = 30027 119910min = 23 1199091max = 34060

1199091min = 35 1198782

119910= 37199578 1198782

1199091

= 39881874 11987821199092

=

31010599 1198622119910

= 18249 1199092max = 28933 119909

2min = 51198622

1199091

= 1892111986221199092

= 17329 1198781199101199091

= 3683802614 1205881199101199091

=

09564 11987811990911199092

= 3387344188 1198781199101199092

= 32395255071205881199101199092

= 09538 and 12058811990911199092

= 09632

The mean squared error of the proposed and the existingestimators is shown in Table 1

5 Conclusion and Future Work

We have developed some ratio product and regressionestimators under maximum and minimum values using twoauxiliary variables The proposed estimators under certainefficiency conditions are shown to be more efficient than theratio product and regression estimators using two auxiliaryvariables The results are shown numerically in Table 1 wherewe observed that the performance of the proposed estimatorsis better than the usual ratio product and the regressionestimators using two auxiliary variablesWe can easily imple-ment the concept of auxiliary variables minimax or maximinestimation of a bounded Poisson (respectively some otherdistribution) mean and censors samples Thus the proposedestimators may be preferred over the existing estimators forthe use of practical applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J Neyman ldquoContributions to the theory of sampling humanpopulationsrdquo Journal of the American Statistical Association vol33 no 201 pp 101ndash116 1938

[2] A K Das and T P Tripathi ldquoSampling strategies for populationmean when the coefficient of variation of an auxiliary characteris knownrdquo Sankhya vol 42 pp 76ndash86 1980

[3] A K Das and T P Tripathi ldquoA class of sampling strategies forpopulation mean using information on mean and variance of

an auxiliary characterrdquo in Proceedings of the Indian StatisticalInstitute Golden Jubilee International Conference on StatisticsApplications and New Directions pp 174ndash181 Calcutta IndiaDecember 1981

[4] L N Upadhyaya and H P Singh ldquoUse of transformed auxiliaryvariable in estimating the finite population meanrdquo BiometricalJournal vol 41 no 5 pp 627ndash636 1999

[5] S Singh ldquoGolden and Silver Jubilee year-2003 of the linearregression estimatorsrdquo in Proceedings of the American StatisticalAssociation Survey Method Section pp 4382ndash4389 AmericanStatistical Association Toronto Canada 2004

[6] B V S Sisodia and V K Dwivedi ldquoA modified ratio estimatorusing coefficient of variation of auxiliary variablerdquo Journal of theIndian Society of Agricultural Statistics vol 33 no 1 pp 13ndash181981

[7] C Kadilar and H Cingi ldquoA new estimator using two auxiliaryvariablesrdquo Applied Mathematics and Computation vol 162 no2 pp 901ndash908 2005

[8] M Khan and J Shabbir ldquoA ratio type estimator for theestimation of population variance using quartiles of an auxiliaryvariablerdquo Journal of Statistics Applications and Probability vol 2no 3 pp 157ndash162 2013

[9] A EMouatasim andAAl-Hossain ldquoReduced gradientmethodfor minimax estimation of a bounded poissonmeanrdquo Journal ofStatistics Advances in Theory and Applications vol 2 no 2 pp185ndash199 2009

[10] Y Al-Hossain ldquoInferences on compound rayleigh parameterswith progressively type-II censored samplesrdquoWorld Academy ofScience Engineering and Technology vol 76 pp 885ndash892 2013

[11] M Khan and J Shabbir ldquoSome improved ratio product andregression estimators of finite population mean when usingminimum and maximum valuesrdquo The Scientific World Journalvol 2013 Article ID 431868 7 pages 2013

[12] C E Sarndal ldquoSample survey theory vs general statistical the-ory estimation of the populationmeanrdquo International StatisticalInstitute vol 40 pp 1ndash12 1972

[13] httpwwwnassusdagovPublicationsAg Statistics1999indexasp

[14] httpwwwnassusdagovPublicationsAg Statistics

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

Journal of Applied Mathematics 5

But when there is positive and negative correlation theregression estimator gives us better result and so for bothcases (positive and negative correlation)wewrite the varianceas

MSE(119884119897119903(119875)

)min

= MSE (1198841198971199032

) minus120579

2 (119873 minus 1)

times ((119910max minus 119910min)

minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min) minus

100381610038161003816100381612057321003816100381610038161003816 (1199092max minus 119909

2min))2

(26)

3 Comparison of Estimators

In this section we have compared the proposed estimatorswith the ratio product and regression estimators and someof their efficiency comparison condition has been carried outunder which the proposed estimators perform better

(i) By (7) and (22)

[MSE (1198841198772

) minusMSE(1198841198771198622

)min

] ge 0 (27)

if

[(119910max minus 119910min) minus 1198771(1199091max minus 119909

1min)

minus1198772(1199092max minus 119909

2min)]2

ge 0

(28)

(ii) By (9) and (23)

[MSE (1198841198752) minusMSE(119884

1198751198622)min

] ge 0 (29)

if

[(119910max minus 119910min) + 1198771(1199091max minus 119909

1min)

+ 1198772(1199092max minus 119909

2min)]2

ge 0

(30)

(iii) By (11) and (26)

[MSE (1198841198971199032

) minusMSE(119884119897119903(119875)

)min

] ge 0 (31)

if

[(119910max minus 119910min) minus10038161003816100381610038161205731

1003816100381610038161003816 (1199091max minus 1199091min)

minus10038161003816100381610038161205732

1003816100381610038161003816 (1199092max minus 1199092min)]

2

ge 0

(32)

From (i) (ii) and (iii) we have observed that the proposedestimators performed better than the other existing estima-tors because the conditions are in the form of a square andgreater than zero which is always true

4 Numerical Illustration

In this section we demonstrate the performance of thesuggested estimators over various other estimators throughfour real data sets The description and the necessary datastatistics of the populations are given as follows

Population 1 (source Agricultural Statistics (1999) [13]Wash-ington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1993

119873 = 69 119899 = 20 1198831

= 4954435 1198832

= 4591072119884 = 4514899 1198782

119910= 37199578 1198782

1199092

= 39881874 1198622119910

=

18249 11986221199091

= 20300 11986221199092

= 18921 1199091max = 38007

1199091min = 32 119910max = 30027 119910min = 23 119909

2max = 340601199092min = 35 1198782

1199091

= 49829270 1205881199101199091

= 09601 1205881199101199092

=

09564 12058811990911199092

= 09729 1198781199101199091

= 4133593285 and 1198781199101199092

=

3683802614

Population 2 (source Agricultural Statistics (1998) [14]Washington US)

119884 season average price per pound during 19961198831 season average price per pound during 1995

1198832 season average price per pound during 1994

119873 = 36 119899 = 12 1198831

= 01856 1198832

= 01708119884 = 02033 119909

1max = 0403 1199091min = 0071 119910max =

0452 119910min = 0101 1199092max = 0334 119909

2min = 00781198782

119910= 0006458 1198782

1199091

= 0005654 11987821199092

= 0004017 1198622119910=

01563 11986221199091

= 01641 11986221199092

= 013757 1205881199101199091

= 087751205881199101199092

= 08577 12058811990911199092

= 08788 1198781199101199091

= 00053 and1198781199101199092

= 00044

Population 3 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1994

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4954435 119883

2= 4230174 119884 =

4514899 1199091max = 38007 119909

1min = 32 1198782119910= 37199578

119910max = 30027 119910min = 23 11987821199091

= 49829270 11987821199092

=

31010599 1199092max = 38933 119909

2min = 5 1198622119910

= 182491198622

1199091

= 20300 11986221199092

= 17329 1198781199101199091

= 41335932851198781199101199092

= 3239525507 11987811990911199092

= 3764276185 1205881199101199091

=

09601 1205881199101199092

= 09538 and 12058811990911199092

= 09576

Population 4 (source Agricultural Statistics (1999) [13]Washington DC)

119884 estimated number of fish caught during 19951198831 estimated number of fish caught during 1993

6 Journal of Applied Mathematics

Table 1 MSE of the existing and the proposed estimators

Estimator Population 1 Population 2 Population 3 Population 4MSE(sdot) MSE(sdot) MSE(sdot) MSE(sdot)

1198841198772

1671738947 00004 1513620253 14406684981198841198752

1216496392 00029 1177239482 11494548321198841198971199032

1254819885 00003 1217975563 1231973764Proposed

1198841198771198622

1293565557 00003 95766257 91198903871198841198751198622

9654413760 00021 8830692238 8616039070119884119897119903119875

9717815401 000025 7538683966 7694770316

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4591072 119883

2= 4230174 119884 =

4514899 119910max = 30027 119910min = 23 1199091max = 34060

1199091min = 35 1198782

119910= 37199578 1198782

1199091

= 39881874 11987821199092

=

31010599 1198622119910

= 18249 1199092max = 28933 119909

2min = 51198622

1199091

= 1892111986221199092

= 17329 1198781199101199091

= 3683802614 1205881199101199091

=

09564 11987811990911199092

= 3387344188 1198781199101199092

= 32395255071205881199101199092

= 09538 and 12058811990911199092

= 09632

The mean squared error of the proposed and the existingestimators is shown in Table 1

5 Conclusion and Future Work

We have developed some ratio product and regressionestimators under maximum and minimum values using twoauxiliary variables The proposed estimators under certainefficiency conditions are shown to be more efficient than theratio product and regression estimators using two auxiliaryvariables The results are shown numerically in Table 1 wherewe observed that the performance of the proposed estimatorsis better than the usual ratio product and the regressionestimators using two auxiliary variablesWe can easily imple-ment the concept of auxiliary variables minimax or maximinestimation of a bounded Poisson (respectively some otherdistribution) mean and censors samples Thus the proposedestimators may be preferred over the existing estimators forthe use of practical applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J Neyman ldquoContributions to the theory of sampling humanpopulationsrdquo Journal of the American Statistical Association vol33 no 201 pp 101ndash116 1938

[2] A K Das and T P Tripathi ldquoSampling strategies for populationmean when the coefficient of variation of an auxiliary characteris knownrdquo Sankhya vol 42 pp 76ndash86 1980

[3] A K Das and T P Tripathi ldquoA class of sampling strategies forpopulation mean using information on mean and variance of

an auxiliary characterrdquo in Proceedings of the Indian StatisticalInstitute Golden Jubilee International Conference on StatisticsApplications and New Directions pp 174ndash181 Calcutta IndiaDecember 1981

[4] L N Upadhyaya and H P Singh ldquoUse of transformed auxiliaryvariable in estimating the finite population meanrdquo BiometricalJournal vol 41 no 5 pp 627ndash636 1999

[5] S Singh ldquoGolden and Silver Jubilee year-2003 of the linearregression estimatorsrdquo in Proceedings of the American StatisticalAssociation Survey Method Section pp 4382ndash4389 AmericanStatistical Association Toronto Canada 2004

[6] B V S Sisodia and V K Dwivedi ldquoA modified ratio estimatorusing coefficient of variation of auxiliary variablerdquo Journal of theIndian Society of Agricultural Statistics vol 33 no 1 pp 13ndash181981

[7] C Kadilar and H Cingi ldquoA new estimator using two auxiliaryvariablesrdquo Applied Mathematics and Computation vol 162 no2 pp 901ndash908 2005

[8] M Khan and J Shabbir ldquoA ratio type estimator for theestimation of population variance using quartiles of an auxiliaryvariablerdquo Journal of Statistics Applications and Probability vol 2no 3 pp 157ndash162 2013

[9] A EMouatasim andAAl-Hossain ldquoReduced gradientmethodfor minimax estimation of a bounded poissonmeanrdquo Journal ofStatistics Advances in Theory and Applications vol 2 no 2 pp185ndash199 2009

[10] Y Al-Hossain ldquoInferences on compound rayleigh parameterswith progressively type-II censored samplesrdquoWorld Academy ofScience Engineering and Technology vol 76 pp 885ndash892 2013

[11] M Khan and J Shabbir ldquoSome improved ratio product andregression estimators of finite population mean when usingminimum and maximum valuesrdquo The Scientific World Journalvol 2013 Article ID 431868 7 pages 2013

[12] C E Sarndal ldquoSample survey theory vs general statistical the-ory estimation of the populationmeanrdquo International StatisticalInstitute vol 40 pp 1ndash12 1972

[13] httpwwwnassusdagovPublicationsAg Statistics1999indexasp

[14] httpwwwnassusdagovPublicationsAg Statistics

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

6 Journal of Applied Mathematics

Table 1 MSE of the existing and the proposed estimators

Estimator Population 1 Population 2 Population 3 Population 4MSE(sdot) MSE(sdot) MSE(sdot) MSE(sdot)

1198841198772

1671738947 00004 1513620253 14406684981198841198752

1216496392 00029 1177239482 11494548321198841198971199032

1254819885 00003 1217975563 1231973764Proposed

1198841198771198622

1293565557 00003 95766257 91198903871198841198751198622

9654413760 00021 8830692238 8616039070119884119897119903119875

9717815401 000025 7538683966 7694770316

1198832 estimated number of fish caught during 1992

119873 = 69 119899 = 20 1198831= 4591072 119883

2= 4230174 119884 =

4514899 119910max = 30027 119910min = 23 1199091max = 34060

1199091min = 35 1198782

119910= 37199578 1198782

1199091

= 39881874 11987821199092

=

31010599 1198622119910

= 18249 1199092max = 28933 119909

2min = 51198622

1199091

= 1892111986221199092

= 17329 1198781199101199091

= 3683802614 1205881199101199091

=

09564 11987811990911199092

= 3387344188 1198781199101199092

= 32395255071205881199101199092

= 09538 and 12058811990911199092

= 09632

The mean squared error of the proposed and the existingestimators is shown in Table 1

5 Conclusion and Future Work

We have developed some ratio product and regressionestimators under maximum and minimum values using twoauxiliary variables The proposed estimators under certainefficiency conditions are shown to be more efficient than theratio product and regression estimators using two auxiliaryvariables The results are shown numerically in Table 1 wherewe observed that the performance of the proposed estimatorsis better than the usual ratio product and the regressionestimators using two auxiliary variablesWe can easily imple-ment the concept of auxiliary variables minimax or maximinestimation of a bounded Poisson (respectively some otherdistribution) mean and censors samples Thus the proposedestimators may be preferred over the existing estimators forthe use of practical applications

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J Neyman ldquoContributions to the theory of sampling humanpopulationsrdquo Journal of the American Statistical Association vol33 no 201 pp 101ndash116 1938

[2] A K Das and T P Tripathi ldquoSampling strategies for populationmean when the coefficient of variation of an auxiliary characteris knownrdquo Sankhya vol 42 pp 76ndash86 1980

[3] A K Das and T P Tripathi ldquoA class of sampling strategies forpopulation mean using information on mean and variance of

an auxiliary characterrdquo in Proceedings of the Indian StatisticalInstitute Golden Jubilee International Conference on StatisticsApplications and New Directions pp 174ndash181 Calcutta IndiaDecember 1981

[4] L N Upadhyaya and H P Singh ldquoUse of transformed auxiliaryvariable in estimating the finite population meanrdquo BiometricalJournal vol 41 no 5 pp 627ndash636 1999

[5] S Singh ldquoGolden and Silver Jubilee year-2003 of the linearregression estimatorsrdquo in Proceedings of the American StatisticalAssociation Survey Method Section pp 4382ndash4389 AmericanStatistical Association Toronto Canada 2004

[6] B V S Sisodia and V K Dwivedi ldquoA modified ratio estimatorusing coefficient of variation of auxiliary variablerdquo Journal of theIndian Society of Agricultural Statistics vol 33 no 1 pp 13ndash181981

[7] C Kadilar and H Cingi ldquoA new estimator using two auxiliaryvariablesrdquo Applied Mathematics and Computation vol 162 no2 pp 901ndash908 2005

[8] M Khan and J Shabbir ldquoA ratio type estimator for theestimation of population variance using quartiles of an auxiliaryvariablerdquo Journal of Statistics Applications and Probability vol 2no 3 pp 157ndash162 2013

[9] A EMouatasim andAAl-Hossain ldquoReduced gradientmethodfor minimax estimation of a bounded poissonmeanrdquo Journal ofStatistics Advances in Theory and Applications vol 2 no 2 pp185ndash199 2009

[10] Y Al-Hossain ldquoInferences on compound rayleigh parameterswith progressively type-II censored samplesrdquoWorld Academy ofScience Engineering and Technology vol 76 pp 885ndash892 2013

[11] M Khan and J Shabbir ldquoSome improved ratio product andregression estimators of finite population mean when usingminimum and maximum valuesrdquo The Scientific World Journalvol 2013 Article ID 431868 7 pages 2013

[12] C E Sarndal ldquoSample survey theory vs general statistical the-ory estimation of the populationmeanrdquo International StatisticalInstitute vol 40 pp 1ndash12 1972

[13] httpwwwnassusdagovPublicationsAg Statistics1999indexasp

[14] httpwwwnassusdagovPublicationsAg Statistics

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