researcharticle the half-logistic lomax distribution for lifetime...

Research ArticleThe Half-Logistic Lomax Distribution for Lifetime Modeling

Masood Anwar and Jawaria Zahoor

Department of Mathematics COMSATS Institute of Information Technology Park Road Chak Shahzad Islamabad Pakistan

Correspondence should be addressed to Masood Anwar masoodanwarcomsatsedupk

Received 7 August 2017 Revised 30 November 2017 Accepted 14 December 2017 Published 1 February 2018

Academic Editor Chin-Shang Li

Copyright copy 2018 Masood Anwar and Jawaria Zahoor 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

We introduce a new two-parameter lifetime distribution called the half-logistic Lomax (HLL) distribution The proposeddistribution is obtained by compounding half-logistic and Lomax distributions We derive some mathematical properties of theproposed distribution such as the survival and hazard rate function quantile function mode median moments and momentgenerating functions mean deviations from mean and median mean residual life function order statistics and entropies Theestimation of parameters is performed by maximum likelihood and the formulas for the elements of the Fisher information matrixare provided A simulation study is run to assess the performance of maximum-likelihood estimators (MLEs) The flexibility andpotentiality of the proposed model are illustrated by means of real and simulated data sets

1 Introduction

The commonly used lifetime distributions (exponentialgamma Weibull Lomax lognormal etc) have a limitedrange of behavior and do not provide adequate fit to com-plex data sets in different sciences Generalizations of thesedistributions offer more flexibility and provide reasonableparametric fits to complex data sets Motivated by the variousapplications of Lomax and half-logistic distributions in areasof income and wealth inequality firm size size of citiesqueuing problems actuarial science medical and biologicalsciences and engineering we propose a two-parametercontinuous lifetime distribution by compounding the half-logistic and the Lomax distribution called half-logistic Lomax(HLL) distribution

The Lomax [1] (or Pareto Type-II) distribution wasintroduced to model business failure data For more detailabout the Lomax distribution we refer the readers to Rady etal [2] Tahir et al [3] and the references therein In literaturethere are several generalizations of the Lomax distributionAbdul-Moniem [4] developed the exponentiated Lomaxdistribution and Al-Awadhi and Ghitany [5] introduced thediscrete PoissonndashLomax distribution by using the Lomax dis-tribution as a mixing distribution for the Poisson parameter

Asgharzadeh et al [6] proposed the Pareto PoissonndashLindelydistribution and Cordeiro et al [7] investigated the gamma-Lomax distribution and studied its properties Ghitany etal [8] and Gupta et al [9] considered the MarshalndashOlkinapproach and extended the Lomax distribution and Lemonteand Cordeiro [10] proposed and studied the McDonald-Lomax the beta Lomax and the Kumaraswamy Lomaxdistributions Other models constitute flexible family ofdistributions in terms of the variates of shapes and hazardfunctions see for example Al-Zahrani and Sagor [11] El-Bassiouny et al [12] Rady et al [2] Kilany [13] and Tahiret al [3] These generalizations of the Lomax distribution areconsidered to be useful life distribution models

The cumulative distribution function (cdf) of Lomaxdistribution is given by

119867(119909 120572 120573) = 1 minus (1 + 120573119909)minus120572 119909 gt 0 (1)

where 120572 gt 0 is a shape parameter and 120573 gt 0 is a scaleparameter The probability density function (pdf) corre-sponding to (1) is

ℎ (119909 120572 120573) = 120572120573 (1 + 120573119909)minus(120572+1) 119909 gt 0 (2)

HindawiJournal of Probability and StatisticsVolume 2018 Article ID 3152807 12 pageshttpsdoiorg10115520183152807

2 Journal of Probability and Statistics

Cordeiro et al [14] define the cdf of the new type I half-logistic-G (TIHL-119866) family of distributions by

119865 (119909 120582 120579) = intminus ln(1minus119866(119909120579))0

2120582119890minus120582119905(1 + 119890minus120582119905)2 119889119905

= 1 minus [1 minus 119866 (119909 120579)]1205821 + [1 minus 119866 (119909 120579)]120582

(3)

where 119866(119909 120579) is the baseline cdf depending on a parametervector 120579 and an additional shape parameter120582 gt 0 As a specialcase if 120582 = 1 then the TIHL-119866 is the half-logistic-119866 (HL-119866)distribution with cdf

119865 (119909 120579) = 119866 (119909 120579)1 + 119866 (119909 120579)

where 119866 (119909 120579) = 1 minus 119866 (119909 120579) (4)

The corresponding pdf to (4) is given by

119891 (119909 120579) = 2119892 (119909 120579)[1 + 119866 (119909 120579)]2 where 119892 (119909) = 119889119866 (119909)119889119909 (5)

This paper aims to provide a new lifetime model witha minimum number of parameters by compounding thehalf-logistic and the Lomax distribution called half-logisticLomax (HLL) distribution The proposed distribution isheavy-tailed and has a decreasing or upside-down bath-tub (or unimodal) shaped hazard rates depending on itsparameters Upside-down bathtub shaped hazard rates arecommon in reliability engineering and survival analysisThe HLL distribution can also be applied in engineering asthe Lomax [1] distribution and can be a useful alternativeto other well-known densities in lifetime applications It isinteresting to note that the HLL distribution is a specialcase of Marshall-OlkinndashLomax distribution introduced byGhitany et al [8] We obtain some mathematical proper-ties of the proposed distribution and parameters of themodel are estimated by the maximum-likelihood estimationmethod

The rest of this paper is organized as follows In Sec-tion 2 we introduce the half-logistic Lomax distributionand provide plots of its density function In Section 3we investigate various mathematical properties of the HLLdistribution including survival and hazard rate functionquantile function moments mean residual life functionmean deviation from the mean and the mean deviationfrom the median entropies and order statistics In Sec-tion 4 estimation of parameters is given by MLE methodand the asymptotic distribution of the estimators is stud-ied via Fisherrsquos information matrix Simulation results onthe behavior of the MLEs are presented in Section 5 Areal data application is conducted in Section 6 Finallyin Section 7 we conclude that the HLL distribution is

0 5 10 15 20 25 30

x

= 01

= 105 = 01

= 15

= 55 = 25

000

002

004

006Den

sity

008

010

012

Figure 1 Plots of the HLL pdf for some parameter values

the best model as compared to other competing mod-els

2 The HLL Distribution

We define the half-logistic-Lomax (HLL) density function byinserting (1) and (2) into (5) So we obtain

119891 (119909 120572 120573) = 2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119909 gt 0 120572 120573 gt 0 (6)

The corresponding cumulative density function (cdf) followsfrom (1) and (4) and is given by

119865 (119909) = [1 minus (1 + 120573119909)minus120572][1 + (1 + 120573119909)minus120572] (7)

Hereafter we will denote a random variable 119883 having pdf(6) by 119883 sim HLL(120572 120573) The limit of the HLL density (6)as 119909 rarr infin is 0 and the limit as 119909 rarr 0 is 1205721205732 Figure 1depicts some of the possible shapes of density (6) for selectedparameter values The mode of density (6) is obtainedfrom solving [119889 log[119891(119909)]119889119909]119909=1199090 = 0 which is givenby

Mode = 1199090 = 1120573 [(120572 minus 1120572 + 1)1120572 minus 1] (8)

3 Mathematical Properties

This section describes mathematical properties of the HLLdistribution

Journal of Probability and Statistics 3

02

04

ℎ(x

)

06

08

10

50 1510 20

x

= 1 = 05 = 1

= 155

= 175

= 195

(a)

2

4

ℎ(x

)

6

8

50 1510 20

x

= 1 = 25 = 5

= 75

= 95

= 125

(b)

Figure 2 Plots of the HLL hrf for some parameter values

31 The Survivor and Hazard Rate Functions Using (6) and(7) the survival function (sf) and hazard rate function (hrf)of119883 are respectively given by

119878 (119909) = 2[1 + (1 + 120573119909)120572]

ℎ (119909) = 120572120573(1 + 120573119909) [1 + (1 + 120573119909)minus120572]

(9)

The limit of ℎ(119909) as 119909 rarr 0+ is 1205721205732 that is ℎ(0) isbounded from below and continuous in its parameters The

limit of ℎ(119909) as 119909 rarr infin is 0 According to Glaser [15] wedetermine the parameter intervals for which the hazard ratefunction of the HLL distribution is decreasing or upside-down bathtub Let

120578 (119909) = minus 119889119889119909 ln119891 (119909)= 120573 [(120572 + 1) minus (120572 minus 1) (1 + 120573119909)minus120572]

(1 + 120573119909) [1 + (1 + 120573119909)minus120572] (10)

Then its first derivative is given by

1205781015840 (119909) = 1205732 [2 (1205722 minus 1) (1 + 120573119909)120572minus2 + (120572 minus 1) (1 + 120573119909)minus2 minus (120572 + 1) (1 + 120573119909)2(120572minus1)][1 + (1 + 120573119909)120572]2 (11)

If 0 lt 120572 le 2 then 1205781015840(119909) lt 0 for all 119909 Then thehazard rate function is decreasing By Glaserrsquos theorem [15]it is sufficient to show that there exists 1199090 gt 0 such that1205781015840(119909) gt 0 for all 119909 isin (0 1199090) 1205781015840(1199090) = 0 implies that 1199090 =(1120573)[(120572 minus 1)(120572 + 1)1120572 minus 1] and 1205781015840(119909) lt 0 for all 119909 gt 1199090which implies unimodal shape of the hazard rate functionFor 120572 gt 2 1205781015840(119909) gt 0 for all 119909 lt 1199090 and 1205781015840(119909) lt 0 for all 119909 gt1199090 Thus the hazard rate function is upside-down bathtub-shaped for 120572 gt 2 Figure 2 illustrates some of the possibleshapes of hazard rate function for selected values of theparameters

32 Quantile and Random Number Generation The cumula-tive distribution function is given by (7) Inverting 119865(119909) = 119906we obtain

119865minus1 (119909) = 1120573 [(1 minus 1199061 + 119906)1120572 minus 1] 0 lt 119906 lt 1 (12)

Equation (12) can be used to simulate the HLL variable

33 Moments The following theorem gives moments ofthe HLL distribution and its mean moments and cumu-lants


Theorem 1 The 119903th moment about the origin of 119883 sim HLL(120572120573) is given by

1205831015840119903 = 119864 [119883119903] = 2120572120573119903infinsum119896=1

119903sum119895=0

(119903119895)

119896 (minus1)119903+119895+119896minus1(119896120572 minus 119895) 119903 = 1 2

(13)

Proof We have

119864 [119883119903] = 2120572120573intinfin0

119909119903 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (14)

Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

119864 [119883119903] = 2120572120573119903 intinfin

1(119910 minus 1)119903 119910minus(120572+1) (1 + 119910minus120572)minus2 119889119910

= 2120572120573119903 intinfin

1

119903sum119895=119900

(119903119895)119910119895minus120572minus1 (minus1)119903+119895

times infinsum119896=1

(minus1)119896minus1 119896 (119910minus120572)119896minus1119889119910 = 2120572120573119903

sdot infinsum119896=1

119903sum119895=119900

(119903119895) (minus1)119903+119895+119896minus1 119896intinfin

1119910119895minus119896120572minus1119889119910 = 2120572120573119903


119903sum119895=119900

(119903119895)

119896 (minus1)119903+119895+119896minus1(119896120572 minus 119895)

(15)

Corollary 2 The first four moments of 119883 sim 119867119871119871(120572 120573) aregiven respectively by

12058310158401 = 119864 [119883] = 2120573infinsum119896=1

(minus1)119896+1 times 1(119896120572 minus 1)

12058310158402 = 119864 [1198832] = 41205732infinsum119896=1

(minus1)119896+1 times 1(119896120572 minus 1) (119896120572 minus 2)

12058310158403 = 119864 [1198833] = 121205733infinsum119896=1

(minus1)119896+1

times 1(119896120572 minus 1) (119896120572 minus 2) (119896120572 minus 3) 12058310158404 = 119864 [1198834] = 481205734

infinsum119896=1

(minus1)119896+1

times 1(119896120572 minus 1) (119896120572 minus 2) (119896120572 minus 3) (119896120572 minus 4)

(16)

Proof Putting 119903 = 1 2 3 4 in (13) yields the desired result

The mean moments (120583119903) and cumulants (120581119903) of 119883 can beobtained by the following relation

120583119903 = 119903sum119895=0

(minus1)119895(119903119895)1205831015840119903minus119895 (12058310158401)119895

120581119903 = 1205831015840119903 minus 119903minus1sum119895=1

(119903 minus 1119895 minus 1)1205811198951205831015840119903minus119895

(17)

Here 1205811 = 12058310158401 We have 1205812 = 12058310158402 minus 120583101584012 1205813 = 12058310158403 minus 31205831015840212058310158401 + 21205831015840131205814 = 12058310158404 minus 41205831015840312058310158401 minus 3120583101584022 + 1212058310158402120583101584012 minus 6120583101584014 and so onThe skewness 1205741 = 1205813120581322 and kurtosis 1205742 = 120581412058122 follow

from the second third and fourth cumulantsThe variance of119883 denoted by 1205902 = 12058310158402 minus 120583101584012 is given by

1205902 = 41205732infinsum119896=1

(minus1)119896+1 1(119896120572 minus 1) (119896120572 minus 2)minus [ 2120573

infinsum119896=1

(minus1)119896+1 1(119896120572 minus 1)]2

(18)

The 119903th descending factorial moment of119883 is

1205831015840(119903) = 119864 [119883(119903)] = 119864 [119883 (119883 minus 1) times sdot sdot sdot times (119883 minus 119903 + 1)]= 119903sum119895=0

119904 (119903 119895) 1205831015840119895 (19)

where

119904 (119903 119895) = 1119895 [ 119889119895119889119909119895 119909(119903)]119909=0

(20)

is the Sterling number of the first kind which counts thenumber of ways to permute a list of 119903 items into 119895 cyclesTheorem 3 The moment generating function of 119883 sim 119867119871119871(120572120573) is given by

119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0

( 119905120573)119894 119896 (minus1)119896minus1(119896120572 minus 119894) 119894 (21)

Proof We have

119864 [119890119905119883] = 2120572120573intinfin0

119890119905119909 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (22)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119864 [119890119905119883] = 2120572119890minus119905120573 intinfin1

119910minus(120572+1)119890119905119910120573 [1 + 119910minus120572]minus2 119889119910 (23)

After simplification we get

119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0



34 Mean Residual Life (MRL) Function The MRL functionis important in reliability survival analysis actuarial scienceseconomics and social sciences for characterizing lifetimedistributions It also plays an important role in repair andreplacement strategies and summarizes the entire residual lifefunction

Theorem 4 TheMRL function of119883 sim 119867119871119871(120572 120573) is given by

119898(119909) = 119864 (119883 minus 119909 | 119883 gt 119909)= 1120573 [1 + (1 + 120573119909)120572] infinsum

119896=1

(minus1)119896minus1 (1 + 120573119909)minus(119896120572minus1)(119896120572 minus 1) (25)

Proof TheMRL function is defined as

119898(119909) = 119864 (119883 minus 119909 | 119883 gt 119909)= 11 minus 119865 (119909) int

infin

119909[1 minus 119865 (119905)] 119889119905 (26)

Therefore we have

119898(119909) = [1 + (1 + 120573119909)120572] intinfin119909

(1 + 120573119905)minus120572[1 + (1 + 120573119905)minus120572]119889119905 (27)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119898(119909) = 1120573 [1 + (1 + 120573119909)120572] intinfin(1+120573119909)

119910minus120572[1 + 119910minus120572]119889119910 (28)

After simplification we get (25)

35 Mean Deviations The mean deviations about mean andmedian are respectively defined by

1205751 (120583) = 2120583119865 (120583) minus 2120583 + 2intinfin120583

119909119891 (119909) 119889119909 (29)

1205752 (119872) = 2119872119865 (119872) minus 119872 minus 120583 + 2intinfin119872

119909119891 (119909) 119889119909 (30)

Theorem 5 The mean deviations about mean and meandeviation about median of119883 sim 119867119871119871(120572 120573) are given by

1205751 (120583) = minus4120583[1 + (1 + 120573120583)120572] + 4120572120573

infinsum119896=1

119896 (minus1)119896minus1

sdot [ 1(119896120572 minus 1) (1 + 120573120583)119896120572minus1 minus

1119896120572 (1 + 120573120583)119896120572]

(31)

1205752 (119872) = 119872[(1 + 120573119872)120572 minus 3][(1 + 120573119872)120572 + 1] minus 2120573

infinsum119896=1

1(119896120572 minus 1) + 4120572120573sdot infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573119872)119896120572]

(32)

Proof Let 119868 = intinfin120583

119909119891(119909)119889119909By inserting pdf of the HLL into (29) and setting 119910 = (1+120573119909) 119889119909 = 119889119910120573 we get119868 = 2120572120573 intinfin

(1+120573120583)(119910minus120572 minus 119910minus(120572+1)) (1 + 119910minus120572)minus2 119889119910 (33)

Using the series representations (1 + 119908)minus2 =suminfin119896=1 119896(minus1)119896minus1119908119896minus1 where 119896 is positive integer we get119868 = 2120572120573

infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(34)

Setting 119868 into (29) and after some manipulations we get thedesired result

Similarly the measure 1205752(119872) can be obtained

36 Entropies The entropy of a random variable 119883 is ameasure of variation of uncertainty Two popular entropiesare the Renyi and Shannon entropies

The Renyi entropy is defined as

119868119877 (120574) = 11 minus 120574 logintinfin0

119891120574 (119909) 119889119909 where 120574 gt 0 120574 = 1

(35)

The Renyi entropy for the HLL distribution is given by

intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903

sdot intinfin0

(1 + 120573119909)minus120574(120572+1) [1 + (1 + 120573119909)minus120572]minus2120574 119889119909(36)

Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903120573 intinfin

1119910minus120574(120572+1) [1 + 119910minus120572]minus2120574 119889119910

= (2120572120573)119903120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1) (37)

Thus we have119868119877 (120574)

= 1(1 minus 120574) log[[(2120572120573)119903

120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1)]] (38)

Another entropymeasure defined by119864minus log[119891(119909)] is knownas the Shannon entropy and plays a similar role as the kurtosismeasure in comparing the shapes of densities and measuringtail heaviness We have119864 minus log [119891 (119909)] = minus log (2120572120573)

+ (120572 + 1) 119864 [log (1 + 120573119909)]+ 2119864 [log 1 + (1 + 120573119909)minus120572]

(39)


By using the binomial series expansion and (2727) ofGradshteyn and Ryzhik [16] it follows that

119864 [log (1 + 120573119909)] = log (4)120572 119864 [log 1 + (1 + 120573119909)minus120572] = [1 minus log (2)]

(40)

Hence the Shannon entropy can be expressed in theform

119864 minus log [119891 (119909)] = log (4)120572 minus log (2120572120573) + 2 (41)

37 Order Statistics Let 1198831 1198832 119883119899 be a random sampleof size 119899 from a distribution with pdf 119891(119909) and cdf 119865(119909) Let1198831119899 1198832119899 119883119899119899 denote the corresponding order statisticsThen the pdf and cdf of119883119903119899 are119891119903119899 (119909) = 1119861 (119903 119899 minus 119903 + 1)119891 (119909) [119865 (119909)]119903minus1

sdot [1 minus 119865 (119909)]119899minus119903= 1119861 (119903 119899 minus 119903 + 1)119891 (119909) 119899minus119903sum

119894=0

(119899 minus 119903119894 ) (minus1)119894 [119865 (119909)]119894+119903minus1

(42)

119865119903119899 (119909) = 119899sum119896=119903

(119899119896) [119865 (119909)]119896 [1 minus 119865 (119909)]119899minus119896

1 le 119903 le 119899(43)

where 119861(119886 119887) is the complete beta function

Theorem 6 Let 119891(119909) and 119865(119909) be the pdf and cdf of 119883 sim119867119871119871(120572 120573) Then the pdf and cdf of119883119903119899 are119891119903119899 (119909) = 1119861 (119903 119899 minus 119903 + 1)

2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2

sdot 119899minus119903sum119894=0

(119899 minus 119903119894 ) (minus1)119894 1 minus (1 + 120573119909)minus120572

1 + (1 + 120573119909)minus120572119894+119903minus1

119865119903119899 (119909) = 119899sum119896=119903

(119899119896)1 minus (1 + 120573119909)minus120572

1 + (1 + 120573119909)minus120572119896

sdot [ 21 + (1 + 120573119909)120572]

119899minus119896

(44)

Proof Inserting (6) and (7) into (44) we get the result

4 Estimation and Asymptotic Distribution

We consider the maximum-likelihood estimation of theparameters 120572 120573 The distributional properties of 120573 areobtained using Fisher information matrix

41 Maximum-Likelihood Estimation Let 1199091 1199092 119909119899 be arandom sample of size 119899 from HLL(120572 120573) distribution Thenthe corresponding likelihood function is

119871 (120572 120573) = (2120572120573)119899 119899prod119894=1

(1 + 120573119909119894)minus(120572+1)[1 + (1 + 120573119909119894)minus120572]2 (45)

The log-likelihood function is given by

ℓ (119909 120572 120573) = 119899 log (2120572120573) minus (120572 + 1) 119899sum119894=1

log (1 + 120573119909119894)

minus 2 119899sum119894=1

log [1 + (1 + 120573119909119894)minus120572] (46)

Calculating the first-order partial derivatives of (46) withrespect to120572120573 and equating them to zero we get the followingnonlinear equations

120597ℓ120597120572 = 119899120572 minus 119899sum119894=1

log (1 + 120573119909119894) + 2 119899sum119894=1

log (1 + 120573119909119894)[1 + (1 + 120573119909119894)120572]= 0

120597ℓ120597120573 = 119899120573 minus (120572 + 1) 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2120572 119899sum119894=1

119909119894(1 + 120573119909119894) [1 + (1 + 120573119909119894)120572] = 0

(47)

To find out the maximum-likelihood estimates (MLEs) of 120572120573 we have to solve the above nonlinear equationsApparentlythere is no closed form solution in 120572 120573 We have to usea numerical technique method such as Newton-Raphsonmethod to obtain the solution

42 Asymptotic Distribution The normal approximation forthe large-sample for the MLE can be used to computeconfidence intervals (CIs) For finding the observed infor-mation matrix of 120572 120573 we compute the second-order partialderivatives of 119897 (46) in the appendix

The variance-covariance matrix may be approximated byΣ = 119868minus1 where 119868 is the observed information matrix SinceΣ involves the parameters 120572 120573 we replace the parameters bythe corresponding MLEs in order to obtain an estimate of Σwhich is denoted by Σ = 119868minus1 where 119868119894119895 = 119868119894119895 when ( 120573)isreplaced by 120572 120573 Using these results an approximate 100(1 minus120599) CI for 120572 120573 respectively is given by

plusmn 1199111205992radicΣ11120573 plusmn 1199111205992radicΣ22

(48)

where 1199111205992 is the upper (1205992)th percentile of the standardnormal distribution


20 40 60 80 100

n

MSE(alpha)MSE(beta)

000

001

002

003

004

005

MSE

of t

he M

LEs (

HLL

)

Figure 3 Plots of the MSE() and MSE(120573) for 120572 = 2 120573 = 1 basedon 10000 replications

5 Simulation Study

The assessment of the maximum-likelihood estimates isbased on a simulation study The following steps were fol-lowed

(1) Generate 5000 samples of size 119899 from (6) The HLLvariables are generated using119883 = (1120573)[((1minus119880)(1+119880))1120572 minus 1] given in (12) where 119880 sim Uniform(0 1)

(2) Compute the MLEs for the 5000 samples say (119894 120573119894)for 119894 = 1 2 5000

(3) Compute the mean square errors (MSE) for eachparameter

We repeat these steps for 119899 = 10 20 100 with 120572 = 2 120573 =1 The comparison is based onMSEs Figure 3 plots theMSEsof the MLEs of two parametersThe assessment based on thissimulation study is that theMSEs for each parameter decreaseto zero with increasing sample size

6 Application

61 Cancer Data We have considered a real data set whichrecords the remission times (in months) of a random sampleof 128 bladder cancer patients reported in [17] We have fittedtheHLL distribution to the data set usingMLE and comparedthe proposed distribution with the following distributions

(i) The Lomax distribution introduced by Lomax [1]with pdf is

119891Lomax (119909) = 120572120573 (1 + 120573119909)minus(120572+1) where 119909 120572 120573 gt 0 (49)

(ii) The McLomax distribution introduced by Lemonteand Cordeiro [10] with pdf is

119891McLomax (119909)= 119888120572120573120572 (120573 + 119909)minus(120572+1)

119861 (119886119888minus1 120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1

sdot [1 minus 1 minus ( 120573120573 + 119909)120572119888]120578 where 119909 120572 120573 119886 119888 gt 0 0 le 120578

(50)

(iii) The beta Lomax (BLomax) distribution is a submodelof McLomax for 119888 = 1 with pdf

119891BLomax (119909) = 120572120573120572 (120573 + 119909)minus(120572+1)119861 (119886 120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1

sdot [1 minus 1 minus ( 120573120573 + 119909)120572]120578 where 119909 120572 120573 119886 gt 0 0 le 120578

(51)

(iv) TheKumaraswamyLomax (KwLomax) distribution isa submodel of McLomax for 119886 = 119888 with pdf

119891KwLomax (119909)= 119886120572120573120572 (120573 + 119909)minus(120572+1)

(120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1


(52)

(v) The Exponentiated Standard Lomax (ESLomax) dis-tribution considered by Gupta et al [18] with pdf is

119891ESLomax (119909) = 120572119886 (1 + 119909)minus(120572+1) 1 minus (1 + 119909)minus120572119886minus1 where 119909 120572 119886 gt 0 (53)

(vi) The Lomax-Logarithmic (LL) distribution intro-duced by Al-Zahrani and Sagor [11] with pdf is

119891LL (119909) = (120578 minus 1) 120572120573 (1 + 120573119909)minus(120572+1)1 minus (1 minus 120578) (1 + 120573119909)minus120572 ln (120578)

where 119909 120572 120573 120578 gt 0(54)

For identification of the shape of the hazard function of thecancer data set we use a graphical method based on the TotalTime on Test (TTT) plot [19] Let 119879 be a random variablewith nonnegative values which represents the survival timeThe empirical TTT curve is constructed by plotting 119879(119903119899) =[sum119903119894=1 119884119894119899 + (119899 minus 119903)119884119903119899](sum119899119894=1 119884119894119899) against 119903119899 where 119903 =1 2 119899 and 119884119894119899 (119894 = 1 2 119899) are the order statisticsof the sample If the empirical TTT transform is straight


Table 1 Descriptive statistics

Mean Median Mode SD Variance Skewness Kurtosis Min Max937 640 50 1051 11043 3287 1548 008 7905

Table 2 MLEs (standard errors in parentheses) and the measures AIC BIC HQIC and CAIC for cancer data

Model Estimates 119888 Measures120572 120573 119886 120578 AIC BIC HQIC CAIC

HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)

diagonal hrf is constant if it is convex hrf is decreasingif it is concave hrf is increasing if it is convex and thenconcave hrf is bathtub and if it is concave and then convexhrf is upside-down bathtub [19] Figure 4 shows the empiricalTTT plot for the data set The shape of the data set isunimodal as its TTT plot is concave and then convex Themodel selection is carried out using Akaike informationcriterion (AIC) the Bayesian information criterion (BIC)the consistent Akaike information criterion (CAIC) and theHannan-Quinn information criterion (HQIC)

AIC = minus2119897 (120579) + 2119902BIC = minus2119897 (120579) + 119902 log (119899)

HQIC = minus2119897 (120579) + 2119902 log (log (119899)) CAIC = minus2119897 (120579) + 2119902119899119899 minus 119902 minus 1

(55)

where 119897(120579) denotes the log-likelihood function evaluated atthe MLEs 119902 is the number of model parameters and 119899 isthe sample size The model with the lowest values for thesestatistics could be chosen as the best model to fit the dataFirst we describe the data set in Table 1 Then we reportthe maximum-likelihood estimators (and the correspondingstandard errors in parentheses) of the parameters and valuesof AIC BIC HQIC and CAIC in Table 2 In Table 2 weprovide some results reported in [10 11] We conclude thatour new model HLL has the smallest AIC BIC HQIC andCAIC values among all fitted models and so it could bechosen as the best model These results are obtained using

00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10

Figure 4 Empirical TTT plot for the cancer data

the AdequacyModel package version 108 available for theprogramming language R [20]

62 Simulated Data Here we used a simulation study tocheck the flexibility of the proposed distribution We havegenerated a sample of size 119899 = 100 from the six distributionsand compared the fit of the HLL distribution with competingmodels

The details of generated samples with the specified valuesof the parameters from the six models are as follows


Table 3 MLEs (standard errors in parentheses) and the measures AIC BIC HQIC and CAIC for simulated data from Lomax distributionwith 120572 = 15 and 120573 = 1Model Estimates 119888 Measures120572 120573 119886 120578 AIC BIC HQIC CAIC

HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)

Table 4MLEs (standard errors in parentheses) and themeasures AIC BIC HQIC andCAIC for simulated data fromMcLomax distributionwith parameters 120572 = 525 120573 = 255 119886 = 145 120578 = 067 and 119888 = 381Model Estimates 119888 Measures120572 120573 119886 120578 AIC BIC HQIC CAIC

HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)

(1) Lomax distribution with 120572 = 15 and 120573 = 1 (seeTable 3)

(2) McLomax distribution with parameters 120572 = 525 120573 =255 119886 = 145 120578 = 067 and 119888 = 381 (see Table 4)(3) BLomax distribution with parameters 120572 = 085 120573 =1255 119886 = 175 and 120578 = 467 (see Table 5)(4) KwLomax distribution with parameters 120572 = 085 120573 =1255 119886 = 175 and 120578 = 467 (see Table 6)(5) ESLomax distribution with parameters 120572 = 25 and119886 = 115 (see Table 7)(6) LL distribution with parameters 120572 = 275 120573 = 045

and 120578 = 075 (see Table 8)The comparison is based upon the formal goodness-of-fit tests given above The values of best fitted model are

highlighted in the tables Based on the goodness-of-fit testswe conclude that the HLL distribution has better fit than theother competing models

7 Concluding Remarks

In this paper we introduce a two-parameterHLL distributionby compounding the half-logistic and the Lomax distribu-tionsThe shape of hazard function of the new compoundingdistribution can be monotonically decreasing or upside-down bathtub (unimodal) Some mathematical and statis-tical properties of the new model including moments andmoment generating functions mean deviations from meanand median quantile function mean residual life functionmode median order statistics and entropies are studied


Table 5 MLEs (standard errors in parentheses) and the measures AIC BIC HQIC and CAIC for simulated data from BLomax distributionwith parameters 120572 = 085 120573 = 1255 119886 = 175 and 120578 = 467Model Estimates 119888 Measures120572 120573 119886 120578 AIC BIC HQIC CAIC

HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)

Table 6MLEs (standard errors in parentheses) and themeasures AIC BIC HQIC and CAIC for simulated data fromKwLomax distributionwith parameters 120572 = 085 120573 = 1255 119886 = 175 and 120578 = 467Model Estimates 119888 Measures120572 120573 119886 120578 AIC BIC HQIC CAIC

HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)

Table 7 MLEs (standard errors in parentheses) and the measures AIC BIC HQIC and CAIC for simulated data from ESLomax distributionwith parameters 120572 = 25 and 119886 = 115Model Estimates 119888 Measures120572 120573 119886 120578 AIC BIC HQIC CAIC

HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)


Table 8 MLEs (standard errors in parentheses) and the measures AIC BIC HQIC and CAIC for simulated data from LL distribution withparameters 120572 = 275 120573 = 045 and 120578 = 075Model Estimates 119888 Measures120572 120573 119886 120578 AIC BIC HQIC CAIC

HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)

We estimate the model parameters by maximum likelihoodand determined the observed informationmatrixWepresenta simulation study to illustrate the performance of MLEsThe flexibility and potentiality of the proposed model areillustrated by means of a real data set We hope that the HLLdistribution may attract wider range of applications in areassuch as engineering survival and lifetime data economicsmeteorology hydrology and others

Appendix

Elements of Observed Information Matrix

The elements of 2 times 2 observed information matrix (119868) aregiven by

11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure

The authors acknowledge that the abstract of the manuscripthad been presented in a conference paper under the linkhttparchivestatuconnedulida17programpdf

Conflicts of Interest

The authors declare that they have no conflicts of interest

Authorsrsquo Contributions

Theauthors have equally contributed to this work All authorsread and approved the final manuscript

References

[1] K S Lomax ldquoBusiness failures another example of the analysisof failure datardquo Journal of the American Statistical Associationvol 49 no 268 pp 847ndash852 1954

[2] E-HA RadyWAHassanein andTA Elhaddad ldquoThepowerLomax distribution with an application to bladder cancer datardquoSpringerPlus vol 5 1838 pages 2016

[3] M H Tahir G M Cordeiro M Mansoor and M ZubairldquoTheWeibull-Lomax distribution properties and applicationsrdquoHacettepe Journal of Mathematics and Statistics vol 44 no 2pp 455ndash474 2015

[4] I B Abdul-Moniem ldquoRecurrence relations for moments oflower generalized order statistics from exponentiated Lomaxdistribution and its characterizationrdquo International Journal ofMathematical Archive vol 3 pp 2144ndash2150 2012

[5] S A Al-Awadhi and M E Ghitany ldquoStatistical properties ofPoisson-Lomax distribution and its application to repeatedaccidents datardquo Journal of Applied Statistical Science vol 10 no4 pp 365ndash372 2001

[6] A Asgharzadeh H S Bakouch and L Esmaeili ldquoParetoPoisson-Lindley distribution with applicationsrdquo Journal ofApplied Statistics vol 40 no 8 pp 1717ndash1734 2013


[7] G M Cordeiro E M Ortega and B z Popovic ldquoThe gamma-Lomax distributionrdquo Journal of Statistical Computation andSimulation vol 85 no 2 pp 305ndash319 2015

[8] M E Ghitany F A Al-Awadhi and L A Alkhalfan ldquoMarshall-Olkin extended Lomax distribution and its application to cen-sored datardquoCommunications in StatisticsmdashTheory andMethodsvol 36 no 9-12 pp 1855ndash1866 2007

[9] R C GuptaM E Ghitany andD K Al-Mutairi ldquoEstimation ofreliability fromMarshall-Olkin extended Lomax distributionsrdquoJournal of Statistical Computation and Simulation vol 80 no7-8 pp 937ndash947 2010

[10] A J Lemonte and G M Cordeiro ldquoAn extended Lomaxdistributionrdquo Statistics vol 47 no 4 pp 800ndash816 2013

[11] B Al-Zahrani and H Sagor ldquoStatistical analysis of the Lomax-logarithmic distributionrdquo Journal of Statistical Computation andSimulation vol 85 no 9 pp 1883ndash1901 2015

[12] A H El-Bassiouny N F Abdo and H S Shahen ldquoExponentiallomax distributionrdquo International Journal of Computer Applica-tions vol 121 no 13 pp 24ndash29 2015

[13] N M Kilany ldquoWeighted Lomax distributionrdquo SpringerPlus vol5 no 1 article no 1862 2016

[14] G M Cordeiro M Alizadeh and P R D Marinho ldquoThetype I half-logistic family of distributionsrdquo Journal of StatisticalComputation and Simulation vol 86 no 4 pp 707ndash728 2016

[15] R E Glaser ldquoBathtub and related failure rate characterizationsrdquoJournal of the American Statistical Association vol 75 no 371pp 667ndash672 1980

[16] I S Gradshteyn and I M Ryzhik Tables of Integrals Series andProducts Academic Press New York NY USA 2007

[17] E T Lee and J W Wang Statistical Methods for Survival DataAnalysis JohnWiley and Sons NewYork NYUSA 3rd edition2003

[18] R C Gupta P L Gupta and R D Gupta ldquoModeling fail-ure time data by Lehman alternativesrdquo Communications inStatisticsmdashTheory andMethods vol 27 no 4 pp 887ndash904 1998

[19] M V Aarset ldquoHow to identify a bathtub hazard raterdquo IEEETransactions on Reliability vol 36 no 1 pp 106ndash108 1987

[20] R Core Team R A Language and Environment for StatisticalComputing R Foundation for Statistical Computing Austria2015 httpswwwR-projectorg

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Cordeiro et al [14] define the cdf of the new type I half-logistic-G (TIHL-119866) family of distributions by

119865 (119909 120582 120579) = intminus ln(1minus119866(119909120579))0

2120582119890minus120582119905(1 + 119890minus120582119905)2 119889119905

= 1 minus [1 minus 119866 (119909 120579)]1205821 + [1 minus 119866 (119909 120579)]120582

(3)

where 119866(119909 120579) is the baseline cdf depending on a parametervector 120579 and an additional shape parameter120582 gt 0 As a specialcase if 120582 = 1 then the TIHL-119866 is the half-logistic-119866 (HL-119866)distribution with cdf

119865 (119909 120579) = 119866 (119909 120579)1 + 119866 (119909 120579)

where 119866 (119909 120579) = 1 minus 119866 (119909 120579) (4)

The corresponding pdf to (4) is given by

119891 (119909 120579) = 2119892 (119909 120579)[1 + 119866 (119909 120579)]2 where 119892 (119909) = 119889119866 (119909)119889119909 (5)

This paper aims to provide a new lifetime model witha minimum number of parameters by compounding thehalf-logistic and the Lomax distribution called half-logisticLomax (HLL) distribution The proposed distribution isheavy-tailed and has a decreasing or upside-down bath-tub (or unimodal) shaped hazard rates depending on itsparameters Upside-down bathtub shaped hazard rates arecommon in reliability engineering and survival analysisThe HLL distribution can also be applied in engineering asthe Lomax [1] distribution and can be a useful alternativeto other well-known densities in lifetime applications It isinteresting to note that the HLL distribution is a specialcase of Marshall-OlkinndashLomax distribution introduced byGhitany et al [8] We obtain some mathematical proper-ties of the proposed distribution and parameters of themodel are estimated by the maximum-likelihood estimationmethod

The rest of this paper is organized as follows In Sec-tion 2 we introduce the half-logistic Lomax distributionand provide plots of its density function In Section 3we investigate various mathematical properties of the HLLdistribution including survival and hazard rate functionquantile function moments mean residual life functionmean deviation from the mean and the mean deviationfrom the median entropies and order statistics In Sec-tion 4 estimation of parameters is given by MLE methodand the asymptotic distribution of the estimators is stud-ied via Fisherrsquos information matrix Simulation results onthe behavior of the MLEs are presented in Section 5 Areal data application is conducted in Section 6 Finallyin Section 7 we conclude that the HLL distribution is

0 5 10 15 20 25 30

x

= 01

= 105 = 01

= 15

= 55 = 25

000

002

004

006Den

sity

008

010

012

Figure 1 Plots of the HLL pdf for some parameter values

the best model as compared to other competing mod-els

2 The HLL Distribution

We define the half-logistic-Lomax (HLL) density function byinserting (1) and (2) into (5) So we obtain

119891 (119909 120572 120573) = 2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119909 gt 0 120572 120573 gt 0 (6)

The corresponding cumulative density function (cdf) followsfrom (1) and (4) and is given by

119865 (119909) = [1 minus (1 + 120573119909)minus120572][1 + (1 + 120573119909)minus120572] (7)

Hereafter we will denote a random variable 119883 having pdf(6) by 119883 sim HLL(120572 120573) The limit of the HLL density (6)as 119909 rarr infin is 0 and the limit as 119909 rarr 0 is 1205721205732 Figure 1depicts some of the possible shapes of density (6) for selectedparameter values The mode of density (6) is obtainedfrom solving [119889 log[119891(119909)]119889119909]119909=1199090 = 0 which is givenby

Mode = 1199090 = 1120573 [(120572 minus 1120572 + 1)1120572 minus 1] (8)

3 Mathematical Properties

This section describes mathematical properties of the HLLdistribution


02

04

ℎ(x

)

06

08

10

50 1510 20

x

= 1 = 05 = 1

= 155

= 175

= 195

(a)

2

4

ℎ(x

)

6

8

50 1510 20

x

= 1 = 25 = 5

= 75

= 95

= 125

(b)



119878 (119909) = 2[1 + (1 + 120573119909)120572]

ℎ (119909) = 120572120573(1 + 120573119909) [1 + (1 + 120573119909)minus120572]

(9)




(1 + 120573119909) [1 + (1 + 120573119909)minus120572] (10)










1205831015840119903 = 119864 [119883119903] = 2120572120573119903infinsum119896=1

119903sum119895=0

(119903119895)


(13)

Proof We have

119864 [119883119903] = 2120572120573intinfin0

119909119903 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (14)

Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

119864 [119883119903] = 2120572120573119903 intinfin


= 2120572120573119903 intinfin

1

119903sum119895=119900





119903sum119895=119900


1119910119895minus119896120572minus1119889119910 = 2120572120573119903


119903sum119895=119900

(119903119895)


(15)


12058310158401 = 119864 [119883] = 2120573infinsum119896=1


12058310158402 = 119864 [1198832] = 41205732infinsum119896=1


12058310158403 = 119864 [1198833] = 121205733infinsum119896=1

(minus1)119896+1


infinsum119896=1

(minus1)119896+1


(16)



120583119903 = 119903sum119895=0

(minus1)119895(119903119895)1205831015840119903minus119895 (12058310158401)119895

120581119903 = 1205831015840119903 minus 119903minus1sum119895=1


(17)



1205902 = 41205732infinsum119896=1


infinsum119896=1

(minus1)119896+1 1(119896120572 minus 1)]2

(18)



119904 (119903 119895) 1205831015840119895 (19)

where

119904 (119903 119895) = 1119895 [ 119889119895119889119909119895 119909(119903)]119909=0

(20)


119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0


Proof We have

119864 [119890119905119883] = 2120572120573intinfin0

119890119905119909 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (22)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119864 [119890119905119883] = 2120572119890minus119905120573 intinfin1



119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0






119896=1




infin

119909[1 minus 119865 (119905)] 119889119905 (26)

Therefore we have

119898(119909) = [1 + (1 + 120573119909)120572] intinfin119909

(1 + 120573119905)minus120572[1 + (1 + 120573119905)minus120572]119889119905 (27)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119898(119909) = 1120573 [1 + (1 + 120573119909)120572] intinfin(1+120573119909)

119910minus120572[1 + 119910minus120572]119889119910 (28)



1205751 (120583) = 2120583119865 (120583) minus 2120583 + 2intinfin120583

119909119891 (119909) 119889119909 (29)


119909119891 (119909) 119889119909 (30)


1205751 (120583) = minus4120583[1 + (1 + 120573120583)120572] + 4120572120573

infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(31)

1205752 (119872) = 119872[(1 + 120573119872)120572 minus 3][(1 + 120573119872)120572 + 1] minus 2120573

infinsum119896=1


119896 (minus1)119896minus1


1119896120572 (1 + 120573119872)119896120572]

(32)





infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(34)






119891120574 (119909) 119889119909 where 120574 gt 0 120574 = 1

(35)


intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903

sdot intinfin0


Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903120573 intinfin


= (2120572120573)119903120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1) (37)

Thus we have119868119877 (120574)

= 1(1 minus 120574) log[[(2120572120573)119903

120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1)]] (38)


+ (120572 + 1) 119864 [log (1 + 120573119909)]+ 2119864 [log 1 + (1 + 120573119909)minus120572]

(39)




(40)





119894=0


(42)

119865119903119899 (119909) = 119899sum119896=119903

(119899119896) [119865 (119909)]119896 [1 minus 119865 (119909)]119899minus119896

1 le 119903 le 119899(43)



2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2

sdot 119899minus119903sum119894=0


1 + (1 + 120573119909)minus120572119894+119903minus1

119865119903119899 (119909) = 119899sum119896=119903

(119899119896)1 minus (1 + 120573119909)minus120572

1 + (1 + 120573119909)minus120572119896

sdot [ 21 + (1 + 120573119909)120572]

119899minus119896

(44)





119871 (120572 120573) = (2120572120573)119899 119899prod119894=1

(1 + 120573119909119894)minus(120572+1)[1 + (1 + 120573119909119894)minus120572]2 (45)


ℓ (119909 120572 120573) = 119899 log (2120572120573) minus (120572 + 1) 119899sum119894=1

log (1 + 120573119909119894)

minus 2 119899sum119894=1

log [1 + (1 + 120573119909119894)minus120572] (46)


120597ℓ120597120572 = 119899120572 minus 119899sum119894=1

log (1 + 120573119909119894) + 2 119899sum119894=1

log (1 + 120573119909119894)[1 + (1 + 120573119909119894)120572]= 0

120597ℓ120597120573 = 119899120573 minus (120572 + 1) 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2120572 119899sum119894=1

119909119894(1 + 120573119909119894) [1 + (1 + 120573119909119894)120572] = 0

(47)





(48)



20 40 60 80 100

n

MSE(alpha)MSE(beta)

000

001

002

003

004

005

MSE

of t

he M

LEs (

HLL

)


5 Simulation Study






6 Application





119891McLomax (119909)= 119888120572120573120572 (120573 + 119909)minus(120572+1)



(50)




(51)


119891KwLomax (119909)= 119886120572120573120572 (120573 + 119909)minus(120572+1)

(120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1


(52)





where 119909 120572 120573 120578 gt 0(54)







HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)




(55)


00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10







HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018









02

04

ℎ(x

)

06

08

10

50 1510 20

x

= 1 = 05 = 1

= 155

= 175

= 195

(a)

2

4

ℎ(x

)

6

8

50 1510 20

x

= 1 = 25 = 5

= 75

= 95

= 125

(b)



119878 (119909) = 2[1 + (1 + 120573119909)120572]

ℎ (119909) = 120572120573(1 + 120573119909) [1 + (1 + 120573119909)minus120572]

(9)




(1 + 120573119909) [1 + (1 + 120573119909)minus120572] (10)










1205831015840119903 = 119864 [119883119903] = 2120572120573119903infinsum119896=1

119903sum119895=0

(119903119895)


(13)

Proof We have

119864 [119883119903] = 2120572120573intinfin0

119909119903 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (14)

Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

119864 [119883119903] = 2120572120573119903 intinfin


= 2120572120573119903 intinfin

1

119903sum119895=119900





119903sum119895=119900


1119910119895minus119896120572minus1119889119910 = 2120572120573119903


119903sum119895=119900

(119903119895)


(15)


12058310158401 = 119864 [119883] = 2120573infinsum119896=1


12058310158402 = 119864 [1198832] = 41205732infinsum119896=1


12058310158403 = 119864 [1198833] = 121205733infinsum119896=1

(minus1)119896+1


infinsum119896=1

(minus1)119896+1


(16)



120583119903 = 119903sum119895=0

(minus1)119895(119903119895)1205831015840119903minus119895 (12058310158401)119895

120581119903 = 1205831015840119903 minus 119903minus1sum119895=1


(17)



1205902 = 41205732infinsum119896=1


infinsum119896=1

(minus1)119896+1 1(119896120572 minus 1)]2

(18)



119904 (119903 119895) 1205831015840119895 (19)

where

119904 (119903 119895) = 1119895 [ 119889119895119889119909119895 119909(119903)]119909=0

(20)


119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0


Proof We have

119864 [119890119905119883] = 2120572120573intinfin0

119890119905119909 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (22)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119864 [119890119905119883] = 2120572119890minus119905120573 intinfin1



119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0






119896=1




infin

119909[1 minus 119865 (119905)] 119889119905 (26)

Therefore we have

119898(119909) = [1 + (1 + 120573119909)120572] intinfin119909

(1 + 120573119905)minus120572[1 + (1 + 120573119905)minus120572]119889119905 (27)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119898(119909) = 1120573 [1 + (1 + 120573119909)120572] intinfin(1+120573119909)

119910minus120572[1 + 119910minus120572]119889119910 (28)



1205751 (120583) = 2120583119865 (120583) minus 2120583 + 2intinfin120583

119909119891 (119909) 119889119909 (29)


119909119891 (119909) 119889119909 (30)


1205751 (120583) = minus4120583[1 + (1 + 120573120583)120572] + 4120572120573

infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(31)

1205752 (119872) = 119872[(1 + 120573119872)120572 minus 3][(1 + 120573119872)120572 + 1] minus 2120573

infinsum119896=1


119896 (minus1)119896minus1


1119896120572 (1 + 120573119872)119896120572]

(32)





infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(34)






119891120574 (119909) 119889119909 where 120574 gt 0 120574 = 1

(35)


intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903

sdot intinfin0


Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903120573 intinfin


= (2120572120573)119903120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1) (37)

Thus we have119868119877 (120574)

= 1(1 minus 120574) log[[(2120572120573)119903

120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1)]] (38)


+ (120572 + 1) 119864 [log (1 + 120573119909)]+ 2119864 [log 1 + (1 + 120573119909)minus120572]

(39)




(40)





119894=0


(42)

119865119903119899 (119909) = 119899sum119896=119903

(119899119896) [119865 (119909)]119896 [1 minus 119865 (119909)]119899minus119896

1 le 119903 le 119899(43)



2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2

sdot 119899minus119903sum119894=0


1 + (1 + 120573119909)minus120572119894+119903minus1

119865119903119899 (119909) = 119899sum119896=119903

(119899119896)1 minus (1 + 120573119909)minus120572

1 + (1 + 120573119909)minus120572119896

sdot [ 21 + (1 + 120573119909)120572]

119899minus119896

(44)





119871 (120572 120573) = (2120572120573)119899 119899prod119894=1

(1 + 120573119909119894)minus(120572+1)[1 + (1 + 120573119909119894)minus120572]2 (45)


ℓ (119909 120572 120573) = 119899 log (2120572120573) minus (120572 + 1) 119899sum119894=1

log (1 + 120573119909119894)

minus 2 119899sum119894=1

log [1 + (1 + 120573119909119894)minus120572] (46)


120597ℓ120597120572 = 119899120572 minus 119899sum119894=1

log (1 + 120573119909119894) + 2 119899sum119894=1

log (1 + 120573119909119894)[1 + (1 + 120573119909119894)120572]= 0

120597ℓ120597120573 = 119899120573 minus (120572 + 1) 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2120572 119899sum119894=1

119909119894(1 + 120573119909119894) [1 + (1 + 120573119909119894)120572] = 0

(47)





(48)



20 40 60 80 100

n

MSE(alpha)MSE(beta)

000

001

002

003

004

005

MSE

of t

he M

LEs (

HLL

)


5 Simulation Study






6 Application





119891McLomax (119909)= 119888120572120573120572 (120573 + 119909)minus(120572+1)



(50)




(51)


119891KwLomax (119909)= 119886120572120573120572 (120573 + 119909)minus(120572+1)

(120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1


(52)





where 119909 120572 120573 120578 gt 0(54)







HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)




(55)


00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10







HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018










1205831015840119903 = 119864 [119883119903] = 2120572120573119903infinsum119896=1

119903sum119895=0

(119903119895)


(13)

Proof We have

119864 [119883119903] = 2120572120573intinfin0

119909119903 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (14)

Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

119864 [119883119903] = 2120572120573119903 intinfin


= 2120572120573119903 intinfin

1

119903sum119895=119900





119903sum119895=119900


1119910119895minus119896120572minus1119889119910 = 2120572120573119903


119903sum119895=119900

(119903119895)


(15)


12058310158401 = 119864 [119883] = 2120573infinsum119896=1


12058310158402 = 119864 [1198832] = 41205732infinsum119896=1


12058310158403 = 119864 [1198833] = 121205733infinsum119896=1

(minus1)119896+1


infinsum119896=1

(minus1)119896+1


(16)



120583119903 = 119903sum119895=0

(minus1)119895(119903119895)1205831015840119903minus119895 (12058310158401)119895

120581119903 = 1205831015840119903 minus 119903minus1sum119895=1


(17)



1205902 = 41205732infinsum119896=1


infinsum119896=1

(minus1)119896+1 1(119896120572 minus 1)]2

(18)



119904 (119903 119895) 1205831015840119895 (19)

where

119904 (119903 119895) = 1119895 [ 119889119895119889119909119895 119909(119903)]119909=0

(20)


119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0


Proof We have

119864 [119890119905119883] = 2120572120573intinfin0

119890119905119909 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2 119889119909 (22)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119864 [119890119905119883] = 2120572119890minus119905120573 intinfin1



119872119883 (119905) = 2120572119890minus119905120573 infinsum119896=1

infinsum119894=0






119896=1




infin

119909[1 minus 119865 (119905)] 119889119905 (26)

Therefore we have

119898(119909) = [1 + (1 + 120573119909)120572] intinfin119909

(1 + 120573119905)minus120572[1 + (1 + 120573119905)minus120572]119889119905 (27)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119898(119909) = 1120573 [1 + (1 + 120573119909)120572] intinfin(1+120573119909)

119910minus120572[1 + 119910minus120572]119889119910 (28)



1205751 (120583) = 2120583119865 (120583) minus 2120583 + 2intinfin120583

119909119891 (119909) 119889119909 (29)


119909119891 (119909) 119889119909 (30)


1205751 (120583) = minus4120583[1 + (1 + 120573120583)120572] + 4120572120573

infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(31)

1205752 (119872) = 119872[(1 + 120573119872)120572 minus 3][(1 + 120573119872)120572 + 1] minus 2120573

infinsum119896=1


119896 (minus1)119896minus1


1119896120572 (1 + 120573119872)119896120572]

(32)





infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(34)






119891120574 (119909) 119889119909 where 120574 gt 0 120574 = 1

(35)


intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903

sdot intinfin0


Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903120573 intinfin


= (2120572120573)119903120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1) (37)

Thus we have119868119877 (120574)

= 1(1 minus 120574) log[[(2120572120573)119903

120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1)]] (38)


+ (120572 + 1) 119864 [log (1 + 120573119909)]+ 2119864 [log 1 + (1 + 120573119909)minus120572]

(39)




(40)





119894=0


(42)

119865119903119899 (119909) = 119899sum119896=119903

(119899119896) [119865 (119909)]119896 [1 minus 119865 (119909)]119899minus119896

1 le 119903 le 119899(43)



2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2

sdot 119899minus119903sum119894=0


1 + (1 + 120573119909)minus120572119894+119903minus1

119865119903119899 (119909) = 119899sum119896=119903

(119899119896)1 minus (1 + 120573119909)minus120572

1 + (1 + 120573119909)minus120572119896

sdot [ 21 + (1 + 120573119909)120572]

119899minus119896

(44)





119871 (120572 120573) = (2120572120573)119899 119899prod119894=1

(1 + 120573119909119894)minus(120572+1)[1 + (1 + 120573119909119894)minus120572]2 (45)


ℓ (119909 120572 120573) = 119899 log (2120572120573) minus (120572 + 1) 119899sum119894=1

log (1 + 120573119909119894)

minus 2 119899sum119894=1

log [1 + (1 + 120573119909119894)minus120572] (46)


120597ℓ120597120572 = 119899120572 minus 119899sum119894=1

log (1 + 120573119909119894) + 2 119899sum119894=1

log (1 + 120573119909119894)[1 + (1 + 120573119909119894)120572]= 0

120597ℓ120597120573 = 119899120573 minus (120572 + 1) 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2120572 119899sum119894=1

119909119894(1 + 120573119909119894) [1 + (1 + 120573119909119894)120572] = 0

(47)





(48)



20 40 60 80 100

n

MSE(alpha)MSE(beta)

000

001

002

003

004

005

MSE

of t

he M

LEs (

HLL

)


5 Simulation Study






6 Application





119891McLomax (119909)= 119888120572120573120572 (120573 + 119909)minus(120572+1)



(50)




(51)


119891KwLomax (119909)= 119886120572120573120572 (120573 + 119909)minus(120572+1)

(120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1


(52)





where 119909 120572 120573 120578 gt 0(54)







HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)




(55)


00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10







HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018












119896=1




infin

119909[1 minus 119865 (119905)] 119889119905 (26)

Therefore we have

119898(119909) = [1 + (1 + 120573119909)120572] intinfin119909

(1 + 120573119905)minus120572[1 + (1 + 120573119905)minus120572]119889119905 (27)

Setting 119910 = (1 + 120573119909)119889119909 = 119889119910120573 yields

119898(119909) = 1120573 [1 + (1 + 120573119909)120572] intinfin(1+120573119909)

119910minus120572[1 + 119910minus120572]119889119910 (28)



1205751 (120583) = 2120583119865 (120583) minus 2120583 + 2intinfin120583

119909119891 (119909) 119889119909 (29)


119909119891 (119909) 119889119909 (30)


1205751 (120583) = minus4120583[1 + (1 + 120573120583)120572] + 4120572120573

infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(31)

1205752 (119872) = 119872[(1 + 120573119872)120572 minus 3][(1 + 120573119872)120572 + 1] minus 2120573

infinsum119896=1


119896 (minus1)119896minus1


1119896120572 (1 + 120573119872)119896120572]

(32)





infinsum119896=1

119896 (minus1)119896minus1


1119896120572 (1 + 120573120583)119896120572]

(34)






119891120574 (119909) 119889119909 where 120574 gt 0 120574 = 1

(35)


intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903

sdot intinfin0


Setting 119910 = (1 + 120573119909) 119889119909 = 119889119910120573 yields

intinfin0

119891120574 (119909) 119889119909 = (2120572120573)119903120573 intinfin


= (2120572120573)119903120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1) (37)

Thus we have119868119877 (120574)

= 1(1 minus 120574) log[[(2120572120573)119903

120573infinsum119895=0

(minus1)119895 (2120574)119895119895 (120574120572 + 119895120572 + 120574 minus 1)]] (38)


+ (120572 + 1) 119864 [log (1 + 120573119909)]+ 2119864 [log 1 + (1 + 120573119909)minus120572]

(39)




(40)





119894=0


(42)

119865119903119899 (119909) = 119899sum119896=119903

(119899119896) [119865 (119909)]119896 [1 minus 119865 (119909)]119899minus119896

1 le 119903 le 119899(43)



2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2

sdot 119899minus119903sum119894=0


1 + (1 + 120573119909)minus120572119894+119903minus1

119865119903119899 (119909) = 119899sum119896=119903

(119899119896)1 minus (1 + 120573119909)minus120572

1 + (1 + 120573119909)minus120572119896

sdot [ 21 + (1 + 120573119909)120572]

119899minus119896

(44)





119871 (120572 120573) = (2120572120573)119899 119899prod119894=1

(1 + 120573119909119894)minus(120572+1)[1 + (1 + 120573119909119894)minus120572]2 (45)


ℓ (119909 120572 120573) = 119899 log (2120572120573) minus (120572 + 1) 119899sum119894=1

log (1 + 120573119909119894)

minus 2 119899sum119894=1

log [1 + (1 + 120573119909119894)minus120572] (46)


120597ℓ120597120572 = 119899120572 minus 119899sum119894=1

log (1 + 120573119909119894) + 2 119899sum119894=1

log (1 + 120573119909119894)[1 + (1 + 120573119909119894)120572]= 0

120597ℓ120597120573 = 119899120573 minus (120572 + 1) 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2120572 119899sum119894=1

119909119894(1 + 120573119909119894) [1 + (1 + 120573119909119894)120572] = 0

(47)





(48)



20 40 60 80 100

n

MSE(alpha)MSE(beta)

000

001

002

003

004

005

MSE

of t

he M

LEs (

HLL

)


5 Simulation Study






6 Application





119891McLomax (119909)= 119888120572120573120572 (120573 + 119909)minus(120572+1)



(50)




(51)


119891KwLomax (119909)= 119886120572120573120572 (120573 + 119909)minus(120572+1)

(120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1


(52)





where 119909 120572 120573 120578 gt 0(54)







HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)




(55)


00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10







HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018











(40)





119894=0


(42)

119865119903119899 (119909) = 119899sum119896=119903

(119899119896) [119865 (119909)]119896 [1 minus 119865 (119909)]119899minus119896

1 le 119903 le 119899(43)



2120572120573 (1 + 120573119909)minus(120572+1)[1 + (1 + 120573119909)minus120572]2

sdot 119899minus119903sum119894=0


1 + (1 + 120573119909)minus120572119894+119903minus1

119865119903119899 (119909) = 119899sum119896=119903

(119899119896)1 minus (1 + 120573119909)minus120572

1 + (1 + 120573119909)minus120572119896

sdot [ 21 + (1 + 120573119909)120572]

119899minus119896

(44)





119871 (120572 120573) = (2120572120573)119899 119899prod119894=1

(1 + 120573119909119894)minus(120572+1)[1 + (1 + 120573119909119894)minus120572]2 (45)


ℓ (119909 120572 120573) = 119899 log (2120572120573) minus (120572 + 1) 119899sum119894=1

log (1 + 120573119909119894)

minus 2 119899sum119894=1

log [1 + (1 + 120573119909119894)minus120572] (46)


120597ℓ120597120572 = 119899120572 minus 119899sum119894=1

log (1 + 120573119909119894) + 2 119899sum119894=1

log (1 + 120573119909119894)[1 + (1 + 120573119909119894)120572]= 0

120597ℓ120597120573 = 119899120573 minus (120572 + 1) 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2120572 119899sum119894=1

119909119894(1 + 120573119909119894) [1 + (1 + 120573119909119894)120572] = 0

(47)





(48)



20 40 60 80 100

n

MSE(alpha)MSE(beta)

000

001

002

003

004

005

MSE

of t

he M

LEs (

HLL

)


5 Simulation Study






6 Application





119891McLomax (119909)= 119888120572120573120572 (120573 + 119909)minus(120572+1)



(50)




(51)


119891KwLomax (119909)= 119886120572120573120572 (120573 + 119909)minus(120572+1)

(120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1


(52)





where 119909 120572 120573 120578 gt 0(54)







HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)




(55)


00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10







HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018









20 40 60 80 100

n

MSE(alpha)MSE(beta)

000

001

002

003

004

005

MSE

of t

he M

LEs (

HLL

)


5 Simulation Study






6 Application





119891McLomax (119909)= 119888120572120573120572 (120573 + 119909)minus(120572+1)



(50)




(51)


119891KwLomax (119909)= 119886120572120573120572 (120573 + 119909)minus(120572+1)

(120578 + 1) 1 minus ( 120573120573 + 119909)120572119886minus1


(52)





where 119909 120572 120573 120578 gt 0(54)







HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)




(55)


00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10







HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018













HLL 52814 00367 82769 83340 83001 82779(20847) (00177)

Lomax 139384 1210225 83167 83737 83398 83176(15368) (142714)

McLomax 08085 112929 15060 41886 21046 82982 84409 83562 83031(3364) (15818) (0234) (25029) (3079)

BLomax 39191 239281 15853 01572 82814 83955 83278 82846(18192) (27338) (0280) (5024)

KwLomax 03911 122973 15162 110323 82788 83929 83252 82821(2386) (17316) (0228) (87144)

ESLomax 10877 46575 85661 86231 83398 85671(0086) (0686)

LL 48696 201652 76938 83088 83944 83436 83107(21317) (183747) (118362)




(55)


00 02 04 06 08 10

in

00

02

T(in

)

04

06

08

10







HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018










HLL 18032 12386 25703 26224 25914 25716(04230) (04764)

Lomax 22441 05337 25739 26260 25950 25752(07434) (02437)

McLomax 27515 23568 09418 minus01016 28189 26290 27593 26817 26354(24395) (02818) (01849) (08005) (3079)

BLomax 14533 22421 09361 06659 26090 27132 26512 26132(81920) (09818) (01699) (00624)

KwLomax 85753 20767 09188 minus07243 26090 27132 26512 26132(07854) (78953) (08728) (71544)

ESLomax 15909 11135 25850 26370 26060 25862(00860) (06861)

LL 26536 03359 05196 25896 26677 26212 25921(03217) (023747) (008362)


HLL 12207 24477 26828 27350 27039 26841(20847) (00177)

Lomax 14243 10314 26749 27270 26960 26761(04009) (04619)

McLomax 141535 32796 08091 minus08197 29511 26695 27997 27222 26758(01364) (53498) (09234) (91029) (40079)

BLomax 19585 37068 06586 03007 26567 27609 26989 26609(60364) (34102) (01077) (40353)

KwLomax 03924 77169 06896 61821 26539 27581 26961 26581(06603) (128494) (00966) (117374)

ESLomax 12764 08217 26494 27015 26704 26506(01748) (01031)

LL 88655 003389 00359 26402 27183 26718 26427(157662) (00695) (00278)









HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018










HLL 244487 00112 53390 53911 53601 53402(137405) (00065)

Lomax 236200 00081 54404 54925 54615 54417(111025) (00038)

McLomax 13135 186449 14939 93587 24534 53496 54799 54023 53560(91629) (731176) (03102) (784737) (38777)

BLomax 20366 939001 17237 138059 53349 54391 53771 53391(265471) (1593791) (02569) (194682)

KwLomax 09258 357112 15236 210737 53305 54347 53727 53347(2386) (17316) (0228) (139842)

ESLomax 13483 50645 56143 56664 56353 56155(01185) (08554)

LL 106838 00392 168989 53587 54369 53904 53612(55325) (00261) (155951)


HLL 32092 01872 45488 46009 45699 45502(10615) (00847)

Lomax 52637 00650 45574 46095 45785 45587(30135) (00436)

McLomax 363053 643913 14745 minus06614 264402 45568 46870 46095 45632(499841) (870837) (02732) (01299) (215904)

BLomax 204375 82358 16963 minus08265 45546 46589 45968 45589(18192) (13201) (03709) (00270)

KwLomax 14823 39164 15331 08043 45573 46615 45995 45615(2386) (70316) (03464) (80144)

ESLomax 13936 26629 45701 46222 45912 45713(01429) (04120)

LL 39058 01236 23957 45748 46530 46065 45773(29870) (01943) (49588)


HLL 29230 14697 7015 7535 7225 7027(08789) (06175)

Lomax 44945 05392 7143 7664 7354 7155(23241) (03345)

McLomax 92546 08577 19039 minus06952 25132 7237 8540 7765 7301(181860) (07067) (09458) (05267) (39407)

BLomax 05806 04735 16416 33194 7061 8104 7483 7104(17619) (03488) (04155) (136281)

KwLomax 24388 05530 16260 minus00069 7064 8107 7486 7107(11386) (06732) (01428) (79604)

ESLomax 34021 13948 6733 7254 6944 6746(04127) (01956)

LL 31584 12813 35394 7286 8068 7603 7312(13691) (14460) (68140)



HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018










HLL 18358 12344 25297 25819 25508 25310(04189) (04553)

Lomax 21811 05669 25322 25843 25533 25334(06717) (02408)

McLomax 07992 12223 09887 16530 19169 25910 27212 26437 25973(28691) (17168) (01874) (101760) (39962)

BLomax 16711 17124 10130 02907 25721 26763 26143 25763(945364) (10299) (01929) (731088)

KwLomax 86899 17774 09917 minus07481 25722 26764 26144 25764(76698) (08502) (02342) (02298)

ESLomax 16418 11489 25410 25931 25621 25423(02057) (01530)

LL 19522 10375 31618 25507 26289 25824 25532(05707) (17363) (111845)


Appendix



11986811 = 12059721198971205971205722 = minus 1198991205722 + 2 119899sum119894=1

(1 + 120573119909119894)120572 [log (1 + 120573119909119894)]2[1 + (1 + 120573119909119894)120572]2

11986812 = 11986821 = 1205972119897120597120572120597120573= minus 119899sum119894=1

119909119894(1 + 120573119909119894)+ 2 119899sum119894=1

119909119894 [1 + (1 + 120573119909119894)120572] + 120572119909119894 log (1 + 120573119909119894)(1 + 120573119909119894)120572+1 [1 + (1 + 120573119909119894)minus120572]2

11986822 = 12059721198971205971205732= minus 1198991205732 + (120572 + 1) 119899sum

119894=1

1199092119894(1 + 120573119909119894)2minus 2120572 119899sum119894=1

1199092119894 [(120572 + 1) (1 + 120573119909119894)120572 + 1](1 + 120573119909119894) [1 + (1 + 120573119909119894)120572]

(A1)

Disclosure






References





























Journal of












Journal of







Volume 2018






Volume 2018






























Journal of












Journal of







Volume 2018






Volume 2018








researcharticle the half-logistic lomax distribution for lifetime...

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