eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 7 Issue 3

A two–parameter weighted Garima distribution with properties and application

Tesfalem Eyob, Rama Shanker
Department of Statistics, Eritrea Institute of Technology, Eritrea
Received: May 03, 2018 | Published: June 13, 2018

Correspondence: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Eritrea, Email

Citation: Eyob T, Shanker R. A two–parameter weighted Garima distribution with properties and application. Biom Biostat Int J. 2018;7(3):234‒242. DOI: 10.15406/bbij.2018.07.00214

Abstract

In this paper a two–parameter weighted Garima distribution which includes one–parameter Garima distribution has been proposed for modeling real lifetime data. Statistical properties of the distribution including shapes of probability density function, moments and moment related measures, hazard rate function, mean residual life function and stochastic orderings have been discussed. The estimation of its parameters has been discussed using the method of maximum likelihood. Application of the proposed distribution has been discussed.

Keywords: garima distribution, moments, hazard rate function, mean residual life function, stochastic ordering, maximum likelihood estimation, goodness of fit

Introduction

Fisher1 firstly introduced the idea of weighted distributions to model ascertainment biases which were later reformulated by Rao2 in a unifying theory for problems where the observations fall in non–experimental, non–replicated and non–random. When a researcher collects observations in the nature according to certain stochastic model, the distribution of the collected observations will not have the original distribution unless every observation has been given an equal chance of being included. Suppose the original observation x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEam aaBaaajuaibaGaaGimaaqcfayabaaaaa@3918@  comes from a population having a probability density function (pdf.), f o (x, θ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaam4BaaqcfayabaGaaiikaiaadIhacaGGSaGaeqiU de3aaSbaaKqbGeaacaaIXaaabeaajuaGcaGGPaaaaa@3F94@ , where θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaabeaaaaa@3944@  may be a parameter vector; and observation x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEaa aa@3781@  is collected according to a probability re–weighted with a weight function w(x, θ 2 )>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Dai aacIcacaWG4bGaaiilaiabeI7aXnaaBaaajuaibaGaaGOmaaqcfaya baGaaiykaiabg6da+iaaicdaaaa@3F97@ , θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIYaaabeaaaaa@3945@  being a new parameter vector, then x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@36F3@  comes from a population having pdf

f( x; θ 1 , θ 2 )=k.w( x; θ 2 ) f o ( x; θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaiaacUdacqaH4oqCdaWgaaqcfasaaiaaigdaaeqa aKqbakaacYcacqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaay jkaiaawMcaaiabg2da9iaadUgacaGGUaGaam4DamaabmaabaGaamiE aiaacUdacqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkai aawMcaaiaaykW7caWGMbWaaSbaaKqbGeaacaWGVbaajuaGbeaadaqa daqaaiaadIhacaGG7aGaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbe aaaiaawIcacaGLPaaaaaa@5714@           (1.1)

where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@  is a normalizing constant. Recall that such types of distributions are known as weighted distributions. A weighted distribution having weight function w( x, θ 2 )=x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Dam aabmaabaGaamiEaiaacYcacqaH4oqCdaWgaaqcfasaaiaaikdaaKqb agqaaaGaayjkaiaawMcaaiabg2da9iaadIhaaaa@4008@  is called length–biased distribution. Extensive discussions on some general probability models leading to weighted probability distributions and their applications and the occurrence of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde Nbambaaaa@3855@  in the problems in sampling have been discussed by Patil & Rao.3,4

Shanker5 proposed a lifetime distribution named Garima distribution for modeling behavioral Science data and defined by its pdf and cumulative distribution function (cdf)

f 1 ( x;θ )= θ θ+2 ( 1+θ+θx ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqabaqcfa4aaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUdehabaGaeq iUdeNaey4kaSIaaGOmaaaadaqadaqaaiaaigdacqGHRaWkcqaH4oqC cqGHRaWkcqaH4oqCcaWG4baacaGLOaGaayzkaaGaamyzamaaCaaabe qcfasaaiabgkHiTiabeI7aXjaadIhaaaqcfaOaaGPaVlaaykW7caGG 7aGaaGPaVlaadIhacqGH+aGpcaaIWaGaaiilaiaaykW7cqaH4oqCcq GH+aGpcaaIWaaaaa@5FCD@              (1.2)

F 1 ( x,θ )=1[ 1+ θx θ+2 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaaiilaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWadaqaai aaigdacqGHRaWkdaWcaaqaaiabeI7aXjaadIhaaeaacqaH4oqCcqGH RaWkcaaIYaaaaaGaay5waiaaw2faaiaadwgadaahaaqabKqbGeaacq GHsislcqaH4oqCcaWG4baaaKqbakaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaacUdacaWG4bGaeyOpa4JaaGimaiaaykW7caaMc8 UaaiilaiaaykW7caaMc8UaeqiUdeNaeyOpa4JaaGimaaaa@66A3@             (1.3)

The first four raw moments (moments about origin) of Garima distribution obtained by Shanker5 are

μ 1 = θ+3 θ( θ+2 ) , μ 2 = 2( θ+4 ) θ 2 ( θ+2 ) , μ 3 = 6( θ+5 ) θ 3 ( θ+2 ) , μ 4 = 24( θ+6 ) θ 4 ( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaabeaajuaGdaahaaqcfasabeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqiUdeNaey4kaSIaaG4maaqaai abeI7aXnaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMca aaaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaacYcacaaMc8UaaG PaVlaaykW7cqaH8oqBdaWgaaqaaiaaikdaaeqaamaaCaaabeqaaiad acUHYaIOaaGaeyypa0ZaaSaaaeaacaaIYaWaaeWaaeaacqaH4oqCcq GHRaWkcaaI0aaacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaeqajuai baGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaikdaaiaawI cacaGLPaaaaaGaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7cqaH8oqB daWgaaqcfasaaiaaiodaaeqaaKqbaoaaCaaajuaibeqaaiadacUHYa IOaaqcfaOaeyypa0ZaaSaaaeaacaaI2aWaaeWaaeaacqaH4oqCcqGH RaWkcaaI1aaacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaeqajuaiba GaaG4maaaajuaGdaqadaqaaiabeI7aXjabgUcaRiaaikdaaiaawIca caGLPaaaaaGaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7cqaH8oqBda WgaaqcfasaaiaaisdaaeqaaKqbaoaaCaaajuaibeqaaiadacUHYaIO aaqcfaOaeyypa0ZaaSaaaeaacaaIYaGaaGinamaabmaabaGaeqiUde Naey4kaSIaaGOnaaGaayjkaiaawMcaaaqaaiabeI7aXnaaCaaabeqc fasaaiaaisdaaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaaca GLOaGaayzkaaaaaaaa@A482@

The central moments (moments about the mean) of Garima distribution obtained by Shanker5 are given by

μ 2 = θ 2 +6θ+7 θ 2 ( θ+2 ) 2 μ 3 = 2( θ 3 +9 θ 2 +21θ+15 ) θ 3 ( θ+2 ) 3 μ 4 = 3( 3 θ 4 +36 θ 3 +134 θ 2 +204θ+111 ) θ 4 ( θ+2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca aMc8UaeqiVd02aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWc aaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG OnaiabeI7aXjabgUcaRiaaiEdaaeaacqaH4oqCdaahaaqabKqbGeaa caaIYaaaaKqbaoaabmaabaGaeqiUdeNaey4kaSIaaGOmaaGaayjkai aawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaqcfayaaiaaykW7cqaH 8oqBdaWgaaqcfasaaiaaiodaaKqbagqaaiabg2da9maalaaabaGaaG OmamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGH RaWkcaaI5aGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRa WkcaaIYaGaaGymaiabeI7aXjabgUcaRiaaigdacaaI1aaacaGLOaGa ayzkaaaabaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGdaqada qaaiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaadaahaaqabKqb GeaacaaIZaaaaaaajuaGcaaMc8UaaGPaVlaaykW7aOqaaKqbakaayk W7cqaH8oqBdaWgaaqcfasaaiaaisdaaKqbagqaaiabg2da9maalaaa baGaaG4mamaabmaabaGaaG4maiabeI7aXnaaCaaabeqcfasaaiaais daaaqcfaOaey4kaSIaaG4maiaaiAdacqaH4oqCdaahaaqabKqbGeaa caaIZaaaaKqbakabgUcaRiaaigdacaaIZaGaaGinaiabeI7aXnaaCa aabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiaaicdacaaI0aGa eqiUdeNaey4kaSIaaGymaiaaigdacaaIXaaacaGLOaGaayzkaaaaba GaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaGdaqadaqaaiabeI7a XjabgUcaRiaaikdaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaI0a aaaaaaaaaa@9DF2@

Statistical properties including shapes for different values of parameter, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistic, Bonferroni and Lorenz curves, Renyi entropy measures and stress–strength reliability of Garima distribution have been discussed in Shanker.5 Estimation of parameter using both the method of moment and the method of maximum likelihood along with application of Garima distribution has been explained in Shanker.5 The suitability, superiority and application of Garima distribution for modeling behavioral science data over exponential and Lindley distribution, introduced by Lindley,6 have also been discussed by Shanker.5 Ghitany et al.,7 have detailed study on statistical properties, estimation of parameter and application of Lindley distribution to model waiting time in a bank and established that Lindley distribution gives much closer fit than exponential distribution. Further, Ghitany et al.,8 introduced a two–parameter weighted Lindley distribution (WLD) defined by its pdf and cdf

f 2 (x;θ,β)= θ β+1 ( θ+β )Γ( β ) x β1 ( 1+x ) e θx ;x>0,θ>0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGOmaaqcfayabaGaaiikaiaadIhacaGG7aGaeqiU deNaaiilaiabek7aIjaacMcacqGH9aqpdaWcaaqaaiabeI7aXnaaCa aabeqcfasaaiabek7aIjabgUcaRiaaigdaaaaajuaGbaWaaeWaaeaa cqaH4oqCcqGHRaWkcqaHYoGyaiaawIcacaGLPaaacqqHtoWrdaqada qaaiabek7aIbGaayjkaiaawMcaaaaacaWG4bWaaWbaaeqajuaibaGa eqOSdiMaeyOeI0IaaGymaaaajuaGdaqadaqaaiaaigdacqGHRaWkca WG4baacaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiab eI7aXjaadIhaaaqcfaOaaGPaVlaaykW7caGG7aGaaGPaVlaaykW7ca WG4bGaeyOpa4JaaGimaiaaykW7caGGSaGaaGPaVlabeI7aXjabg6da +iaaicdacaGGSaGaaGPaVlabek7aIjabg6da+iaaicdaaaa@7695@        (1.4)

F 2 (x;θ,β)=1 ( θ+β )Γ( β,θx )+ ( θx ) β e θx ( θ+β )Γ( β ) ;x>0,θ>0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGOmaaqcfayabaGaaiikaiaadIhacaGG7aGaeqiU deNaaiilaiabek7aIjaacMcacqGH9aqpcaaIXaGaeyOeI0YaaSaaae aadaqadaqaaiabeI7aXjabgUcaRiabek7aIbGaayjkaiaawMcaaiab fo5ahnaabmaabaGaeqOSdiMaaiilaiabeI7aXjaadIhaaiaawIcaca GLPaaacqGHRaWkdaqadaqaaiabeI7aXjaadIhaaiaawIcacaGLPaaa daahaaqabeaacqaHYoGyaaGaamyzamaaCaaabeqcfasaaiabgkHiTi abeI7aXjaadIhaaaaajuaGbaWaaeWaaeaacqaH4oqCcqGHRaWkcqaH YoGyaiaawIcacaGLPaaacqqHtoWrdaqadaqaaiabek7aIbGaayjkai aawMcaaaaacaaMc8UaaGPaVlaacUdacaaMc8UaaGPaVlaaykW7caWG 4bGaeyOpa4JaaGimaiaacYcacaaMc8UaeqiUdeNaeyOpa4JaaGimai aacYcacaaMc8UaeqOSdiMaeyOpa4JaaGimaaaa@7D9C@         (1.5)

where Γ( β,θx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeu4KdC 0aaeWaaeaacqaHYoGycaGGSaGaeqiUdeNaamiEaaGaayjkaiaawMca aaaa@3E79@  is the upper incomplete gamma function defined as

Γ( β,z )= z e y y β1 dy;y0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeu4KdC 0aaeWaaeaacqaHYoGycaGGSaGaamOEaaGaayjkaiaawMcaaiabg2da 9maapehabaGaamyzamaaCaaabeqaaiabgkHiTiaadMhaaaaajuaiba GaamOEaaqaaiabg6HiLcqcfaOaey4kIipacaWG5bWaaWbaaeqajuai baGaeqOSdiMaeyOeI0IaaGymaaaajuaGcaWGKbGaamyEaiaacUdaca WG5bGaeyyzImRaaGimaiaacYcacaaMc8UaeqOSdiMaeyOpa4JaaGim aaaa@5726@        (1.6)

It can be easily shown that Lindley distribution is a particular case of WLD at β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGymaaaa@39E6@ . Shanker et al.,9 proposed an extension of WLD named a three–parameter weighted Lindley distribution which includes Lindley distribution and WLD as particular cases.

Shanker10 has obtained discrete Poisson–Garima distribution (PGD), discussed its statistical properties, estimation of parameter and applications for count data from biological sciences. Shanker & Shukla11,12 have also proposed size–biased Poisson–Garima distribution (SBPGD) and Zero–truncated Poisson–Garima distribution (ZTPGD), along with estimation of their parameter and applications for data which structurally excludes zero counts.

The organizations of the rest of the paper are as follows: Section 2 deals with two–parameter weighted Garima distribution (WGD) which includes one–parameter Garima distribution proposed by Shanker5 along with the shapes of the pdf and the cdf of WGD. Section 3 deals with statistical constants and associated measures of WGD including coefficient of variation, skewness, kurtosis, index of dispersion. Section 4 deals with the reliability properties of WGD including hazard rate function, mean residual life function and stochastic ordering. Section 5 deals with the maximum likelihood method for the estimation of parameters of the distribution. Section 6 deals with the goodness of fit of the proposed distribution with a real lifetime data over other one parameter and two–parameter lifetime distributions. Finally, the conclusions of the paper have been presented in section 7.

Weighted Garima distribution

The pdf of the weighted Garima distribution (WGD) can be expressed as

f 3 ( x;θ,β )=K x β1 f o ( x;θ );x>0,θ>0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXjaacYcacqaHYoGyaiaawIcacaGLPaaacqGH9aqpcaWGlbGaaG PaVlaadIhadaahaaqabKqbGeaacqaHYoGycqGHsislcaaIXaaaaKqb akaadAgadaWgaaqcfasaaiaad+gaaKqbagqaamaabmaabaGaamiEai aacUdacqaH4oqCaiaawIcacaGLPaaacaaMc8Uaai4oaiaaykW7caaM c8UaamiEaiabg6da+iaaicdacaGGSaGaaGPaVlabeI7aXjabg6da+i aaicdacaGGSaGaaGPaVlabek7aIjabg6da+iaaicdaaaa@6401@  (2.1)

where, K is the normalizing constant and f o (x;θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaam4BaaqcfayabaGaaiikaiaadIhacaGG7aGaeqiU deNaaiykaiaaykW7aaa@3F96@ is the pdf of Garima distribution given in (1.2). Thus the pdf of WGD can be obtained as

f 3 ( x;θ,β )= θ β ( θ+β+1 ) x β1 Γ( β ) ( 1+θ+θx ) e θx ;x>0,θ>0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXjaacYcacqaHYoGyaiaawIcacaGLPaaacqGH9aqpdaWcaaqaai abeI7aXnaaCaaabeqcfasaaiabek7aIbaaaKqbagaadaqadaqaaiab eI7aXjabgUcaRiabek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaaaa GaaGPaVlaaykW7daWcaaqaaiaadIhadaahaaqabKqbGeaacqaHYoGy cqGHsislcaaIXaaaaaqcfayaaiabfo5ahnaabmaabaGaeqOSdigaca GLOaGaayzkaaaaamaabmaabaGaaGymaiabgUcaRiabeI7aXjabgUca RiabeI7aXjaadIhaaiaawIcacaGLPaaacaWGLbWaaWbaaeqajuaiba GaeyOeI0IaeqiUdeNaamiEaaaajuaGcaaMc8UaaGPaVlaacUdacaaM c8UaaGPaVlaadIhacqGH+aGpcaaIWaGaaiilaiaaykW7cqaH4oqCcq GH+aGpcaaIWaGaaiilaiaaykW7caaMc8UaeqOSdiMaeyOpa4JaaGim aaaa@7E3A@  (2.2)

where

Γ( β )= 0 e y y β1 dy;y>0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeu4KdC 0aaeWaaeaacqaHYoGyaiaawIcacaGLPaaacqGH9aqpdaWdXbqaaiaa dwgadaahaaqabKqbGeaacqGHsislcaWG5baaaaqaaiaaicdaaeaacq GHEisPaKqbakabgUIiYdGaamyEamaaCaaabeqcfasaaiabek7aIjab gkHiTiaaigdaaaqcfaOaamizaiaadMhacaGG7aGaamyEaiabg6da+i aaicdacaGGSaGaeqOSdiMaeyOpa4JaaGimaaaa@52E9@  is the complete gamma function.

We say that X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@  follows WGD with parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  if its pdf is given by (2.2) and we denote it by X~WGD( θ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai aac6hacaqGxbGaae4raiaabseacaaMc8+aaeWaaeaacqaH4oqCcaGG SaGaeqOSdigacaGLOaGaayzkaaaaaa@41E9@ . It can be easily verified that Garima distribution and size–biased Garima distribution (SBGD) are particular cases of WGD at β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGymaaaa@39E6@  and β=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGOmaaaa@39E7@ , respectively. The behavior of the pdf of WGD for different combinations of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are shown in Figure 1.

Figure 1 Behavior of the pdf of WGD for different combinations of the parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo Gyaaa@3826@ .

The cdf of the WGD can be expressed as

F 3 ( x;θ,β )=1 ( θx ) β e θx +( θ+β+1 )Γ( β,θx ) ( θ+β+1 )Γ( β ) ;x>0,θ>0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXjaacYcacqaHYoGyaiaawIcacaGLPaaacqGH9aqpcaaIXaGaey OeI0YaaSaaaeaadaqadaqaaiabeI7aXjaadIhaaiaawIcacaGLPaaa daahaaqabeaacqaHYoGyaaGaamyzamaaCaaabeqcfasaaiabgkHiTi abeI7aXjaadIhaaaqcfaOaey4kaSYaaeWaaeaacqaH4oqCcqGHRaWk cqaHYoGycqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeu4KdC0aaeWaae aacqaHYoGycaGGSaGaeqiUdeNaamiEaaGaayjkaiaawMcaaaqaamaa bmaabaGaeqiUdeNaey4kaSIaeqOSdiMaey4kaSIaaGymaaGaayjkai aawMcaaiabfo5ahnaabmaabaGaeqOSdigacaGLOaGaayzkaaaaaiaa ykW7caGG7aGaamiEaiabg6da+iaaicdacaGGSaGaeqiUdeNaeyOpa4 JaaGimaiaacYcacqaHYoGycqGH+aGpcaaIWaaaaa@77C5@     (2.3)

where Γ( β,θx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHto WrjuaGdaqadaGcbaqcLbsacqaHYoGycaGGSaGaeqiUdeNaamiEaaGc caGLOaGaayzkaaaaaa@3FAB@  is the upper incomplete gamma function defined in (1.6). Behavior of the cdf of the WGD for different combinations of the parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are shown in Figure 2.

Figure 2 Behavior of the cdf of WGD for different combinations of the parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycaGGUaaaaa@38D8@

Statistical constants and related measures

The rth raw moments (moment about origin), μ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaGPaVl abeY7aTnaaBaaajuaibaGaamOCaaqabaqcfa4aaWbaaKqbGeqabaGa mai4gkdiIcaaaaa@3ED1@ ,of WGD (2.2) can be derived as

μ r =E( X r )= 0 x r f 2 ( x;θ,β )dx= 0 x r x β1 ( θ+β+1 ) θ β Γ( β ) ( 1+θ+θx ) e θx dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaWGYbaabeaajuaGdaahaaqcfasabeaacWaGGBOm GikaaKqbakabg2da9iaadweadaqadaqaaiaadIfadaahaaqabKqbGe aacaWGYbaaaaqcfaOaayjkaiaawMcaaiabg2da9maapehabaGaamiE amaaCaaabeqcfasaaiaadkhaaaqcfaOaamOzamaaBaaajuaibaGaaG OmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiabeI7aXjaacYcacqaH YoGyaiaawIcacaGLPaaacaWGKbGaamiEaiabg2da9maapehabaGaam iEamaaCaaabeqcfasaaiaadkhaaaqcfa4aaSaaaeaacaWG4bWaaWba aeqajuaibaGaeqOSdiMaeyOeI0IaaGymaaaaaKqbagaadaqadaqaai abeI7aXjabgUcaRiabek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaa aaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacqaHYoGyaaaajuaGba Gaeu4KdC0aaeWaaeaacqaHYoGyaiaawIcacaGLPaaaaaWaaeWaaeaa caaIXaGaey4kaSIaeqiUdeNaey4kaSIaeqiUdeNaamiEaaGaayjkai aawMcaaiaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqCcaWG4baa aKqbakaadsgacaWG4baajuaibaGaaGimaaqaaiabg6HiLcqcfaOaey 4kIipacaaMc8UaaGPaVdqcfasaaiaaicdaaeaacqGHEisPaKqbakab gUIiYdaaaa@89F4@

= ( θ+β+r+1 )Γ( β+r ) θ r ( θ+β+1 )Γ( β ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaadaqadaqaaiabeI7aXjabgUcaRiabek7aIjabgUcaRiaa dkhacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeu4KdC0aaeWaaeaacq aHYoGycqGHRaWkcaWGYbaacaGLOaGaayzkaaaabaGaeqiUde3aaWba aeqajuaibaGaamOCaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiabek 7aIjabgUcaRiaaigdaaiaawIcacaGLPaaacqqHtoWrdaqadaqaaiab ek7aIbGaayjkaiaawMcaaaaacaaMc8UaaGPaVlaaykW7caGG7aGaaG PaVlaadkhacqGH9aqpcaaIXaGaaiilaiaaykW7caaIYaGaaiilaiaa ykW7caaIZaGaaiilaiaaykW7caGGUaGaaGPaVlaac6cacaaMc8Uaai Olaaaa@6DB2@               (3.1)

Thus the first four raw moments of WGD are obtained as

μ 1 = β( θ+β+2 ) θ( θ+β+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaabeaajuaGdaahaaqcfasabeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqOSdi2aaeWaaeaacqaH4oqCcq GHRaWkcqaHYoGycqGHRaWkcaaIYaaacaGLOaGaayzkaaaabaGaeqiU de3aaeWaaeaacqaH4oqCcqGHRaWkcqaHYoGycqGHRaWkcaaIXaaaca GLOaGaayzkaaaaaaaa@50C4@

μ 2 = β( β+1 )( θ+β+3 ) θ 2 ( θ+β+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaabeaajuaGdaahaaqcfasabeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqOSdi2aaeWaaeaacqaHYoGycq GHRaWkcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCcqGHRaWk cqaHYoGycqGHRaWkcaaIZaaacaGLOaGaayzkaaaabaGaeqiUde3aaW baaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiab ek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaaaaaaaa@5727@

μ 3 = β( β+1 )( β+2 )( θ+β+4 ) θ 3 ( θ+β+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaabeaajuaGdaahaaqcfasabeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqOSdi2aaeWaaeaacqaHYoGycq GHRaWkcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacqaHYoGycqGHRaWk caaIYaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCcqGHRaWkcqaHYo GycqGHRaWkcaaI0aaacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaeqa juaibaGaaG4maaaajuaGdaqadaqaaiabeI7aXjabgUcaRiabek7aIj abgUcaRiaaigdaaiaawIcacaGLPaaaaaaaaa@5BF2@

μ 4 = β( β+1 )( β+2 )( β+3 )( θ+β+5 ) θ 4 ( θ+β+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaabeaajuaGdaahaaqcfasabeaacWaGGBOm GikaaKqbakabg2da9maalaaabaGaeqOSdi2aaeWaaeaacqaHYoGycq GHRaWkcaaIXaaacaGLOaGaayzkaaWaaeWaaeaacqaHYoGycqGHRaWk caaIYaaacaGLOaGaayzkaaWaaeWaaeaacqaHYoGycqGHRaWkcaaIZa aacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCcqGHRaWkcqaHYoGycqGH RaWkcaaI1aaacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaeqajuaiba GaaGinaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiabek7aIjabgUca RiaaigdaaiaawIcacaGLPaaaaaaaaa@60BE@

Using relationship μ r =E ( X μ 1 ) r = k=0 r ( r k ) μ k ( μ 1 ) rk MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaWGYbaajuaGbeaacqGH9aqpcaWGfbWaaeWaaeaa caWGybGaeyOeI0IaeqiVd02aaSbaaKqbGeaacaaIXaaabeaajuaGda ahaaqcfasabeaacWaGGBOmGikaaaqcfaOaayjkaiaawMcaamaaCaaa beqcfasaaiaadkhaaaqcfaOaeyypa0ZaaabCaeaadaqadaabaeqaba GaamOCaaqaaiaadUgaaaGaayjkaiaawMcaaaqcfasaaiaadUgacqGH 9aqpcaaIWaaabaGaamOCaaqcfaOaeyyeIuoacaaMc8UaeqiVd02aaS baaKqbGeaacaWGRbaabeaajuaGdaahaaqcfasabeaacWaGGBOmGika aKqbakaaykW7daqadaqaaiabgkHiTiabeY7aTnaaBaaajuaibaGaaG ymaaqabaqcfa4aaWbaaKqbGeqabaGamai4gkdiIcaaaKqbakaawIca caGLPaaadaahaaqabKqbGeaacaWGYbGaeyOeI0Iaam4Aaaaaaaa@6A31@  between central moments and raw moments, the central moments of WGD are

μ 2 = β{ θ 2 +( 2β+4 )θ+( β 2 +3β+3 ) } θ 2 ( θ+β+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabek7a InaacmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRa WkdaqadaqaaiaaikdacqaHYoGycqGHRaWkcaaI0aaacaGLOaGaayzk aaGaeqiUdeNaey4kaSYaaeWaaeaacqaHYoGydaahaaqabKqbGeaaca aIYaaaaKqbakabgUcaRiaaiodacqaHYoGycqGHRaWkcaaIZaaacaGL OaGaayzkaaaacaGL7bGaayzFaaaabaGaeqiUde3aaWbaaeqajuaiba GaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgUcaRiabek7aIjabgUca RiaaigdaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaaaaaaaa a@6069@

μ 3 = 2β{ θ 3 +( 3β+6 ) θ 2 +( 3 β 2 +9β+9 )θ+( β 3 +4 β 2 +6β+4 ) } θ 3 ( θ+β+1 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIZaaajuaGbeaacqGH9aqpdaWcaaqaaiaaikda cqaHYoGydaGadaqaaiabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfa Oaey4kaSYaaeWaaeaacaaIZaGaeqOSdiMaey4kaSIaaGOnaaGaayjk aiaawMcaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaS YaaeWaaeaacaaIZaGaeqOSdi2aaWbaaeqajuaibaGaaGOmaaaajuaG cqGHRaWkcaaI5aGaeqOSdiMaey4kaSIaaGyoaaGaayjkaiaawMcaai abeI7aXjabgUcaRmaabmaabaGaeqOSdi2aaWbaaeqajuaibaGaaG4m aaaajuaGcqGHRaWkcaaI0aGaeqOSdi2aaWbaaeqajuaqbaGaaGOmaa aajuaGcqGHRaWkcaaI2aGaeqOSdiMaey4kaSIaaGinaaGaayjkaiaa wMcaaaGaay5Eaiaaw2haaaqaaiabeI7aXnaaCaaabeqcfasaaiaaio daaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcqaHYoGycqGHRaWkcaaI XaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaG4maaaaaaaaaa@74CA@

μ 4 = 3β{ ( β+2 ) θ 4 +( 4 β 2 +16β+16 ) θ 3 +( 6 β 3 +34 β 2 +58β+36 ) θ 2 +( 4 β 4 +28 β 3 +68 β 2 +72β+32 )θ +( β 5 +8 β 4 +25 β 3 +38 β 2 +29β+10 ) } θ 4 ( θ+β+1 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaI0aaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda cqaHYoGydaGadaabaeqabaWaaeWaaeaacqaHYoGycqGHRaWkcaaIYa aacaGLOaGaayzkaaGaeqiUde3aaWbaaeqajuaibaGaaGinaaaajuaG cqGHRaWkdaqadaqaaiaaisdacqaHYoGydaahaaqabKqbGeaacaaIYa aaaKqbakabgUcaRiaaigdacaaI2aGaeqOSdiMaey4kaSIaaGymaiaa iAdaaiaawIcacaGLPaaacqaH4oqCdaahaaqabKqbGeaacaaIZaaaaK qbakabgUcaRmaabmaabaGaaGOnaiabek7aInaaCaaabeqcfasaaiaa iodaaaqcfaOaey4kaSIaaG4maiaaisdacqaHYoGydaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaiwdacaaI4aGaeqOSdiMaey4kaSIa aG4maiaaiAdaaiaawIcacaGLPaaacqaH4oqCdaahaaqabKqbGeaaca aIYaaaaaqcfayaaiabgUcaRmaabmaabaGaaGinaiabek7aInaaCaaa beqcfasaaiaaisdaaaqcfaOaey4kaSIaaGOmaiaaiIdacqaHYoGyda ahaaqabKqbGeaacaaIZaaaaKqbakabgUcaRiaaiAdacaaI4aGaeqOS di2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI3aGaaGOmai abek7aIjabgUcaRiaaiodacaaIYaaacaGLOaGaayzkaaGaeqiUdeha baGaey4kaSYaaeWaaeaacqaHYoGydaahaaqabKqbGeaacaaI1aaaaK qbakabgUcaRiaaiIdacqaHYoGydaahaaqabKqbGeaacaaI0aaaaKqb akabgUcaRiaaikdacaaI1aGaeqOSdi2aaWbaaeqajuaibaGaaG4maa aajuaGcqGHRaWkcaaIZaGaaGioaiabek7aInaaCaaabeqcfasaaiaa ikdaaaqcfaOaey4kaSIaaGOmaiaaiMdacqaHYoGycqGHRaWkcaaIXa GaaGimaaGaayjkaiaawMcaaaaacaGL7bGaayzFaaaabaGaeqiUde3a aWbaaeqajuaibaGaaGinaaaajuaGdaqadaqaaiabeI7aXjabgUcaRi abek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaadaahaaqabKqbGeaa caaI0aaaaaaaaaa@AEE7@

It can be verified that at β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGymaaaa@39E6@ , the moments about origin and the moments about mean of WGD reduces to the corresponding moments of Garima distribution.

The coefficient variation (C.V.) coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B47@  and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@  of WGD are thus given as

C.V= σ μ 1 = θ 2 +( 2β+4 )θ+( β 2 +3β+3 ) β ( θ+β+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacuaH8oqBgaqb amaaBaaajuaibaGaaGymaaqabaaaaKqbakabg2da9maalaaabaWaaO aaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRmaa bmaabaGaaGOmaiabek7aIjabgUcaRiaaisdaaiaawIcacaGLPaaacq aH4oqCcqGHRaWkdaqadaqaaiabek7aInaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSIaaG4maiabek7aIjabgUcaRiaaiodaaiaawIcaca GLPaaaaeqaaaqaamaakaaabaGaeqOSdigabeaadaqadaqaaiabeI7a XjabgUcaRiabek7aIjabgUcaRiaaikdaaiaawIcacaGLPaaaaaaaaa@5F36@

β 1 = μ 3 μ 2 3/2 = 2{ θ 3 +( 3β+6 ) θ 2 +( 3 β 2 +9β+9 )θ+( β 3 +4 β 2 +6β+4 ) } β { θ 2 +( 2β+4 )θ+( β 2 +3β+3 ) } 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaeyypa0Za aSaaaeaacqaH8oqBdaWgaaqcfasaaiaaiodaaKqbagqaaaqaaiabeY 7aTnaaDaaajuaibaGaaGOmaaqaaKqbaoaalyaajuaibaGaaG4maaqa aiaaikdaaaaaaaaajuaGcqGH9aqpdaWcaaqaaiaaikdadaGadaqaai abeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaey4kaSYaaeWaaeaa caaIZaGaeqOSdiMaey4kaSIaaGOnaaGaayjkaiaawMcaaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSYaaeWaaeaacaaIZaGa eqOSdi2aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI5aGaeq OSdiMaey4kaSIaaGyoaaGaayjkaiaawMcaaiabeI7aXjabgUcaRmaa bmaabaGaeqOSdi2aaWbaaeqabaGaaG4maaaacqGHRaWkcaaI0aGaeq OSdi2aaWbaaeqabaGaaGOmaaaacqGHRaWkcaaI2aGaeqOSdiMaey4k aSIaaGinaaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaqaamaakaaaba GaeqOSdigabeaadaGadaqaaiabeI7aXnaaCaaabeqcfasaaiaaikda aaqcfaOaey4kaSYaaeWaaeaacaaIYaGaeqOSdiMaey4kaSIaaGinaa GaayjkaiaawMcaaiabeI7aXjabgUcaRmaabmaabaGaeqOSdi2aaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIZaGaeqOSdiMaey4kaS IaaG4maaGaayjkaiaawMcaaaGaay5Eaiaaw2haamaaCaaabeqaamaa lyaabaGaaG4maaqaaiaaikdaaaaaaaaaaaa@8B54@

β 2 = μ 4 μ 2 2 = 3{ ( β+2 ) θ 4 +( 4 β 2 +16β+16 ) θ 3 +( 6 β 3 +34 β 2 +58β+36 ) θ 2 +( 4 β 4 +28 β 3 +68 β 2 +72β+32 )θ +( β 5 +8 β 4 +25 β 3 +38 β 2 +29β+10 ) } β { θ 2 +( 2β+4 )θ+( β 2 +3β+3 ) } 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeY7a TnaaBaaajuaibaGaaGinaaqcfayabaaabaGaeqiVd02aa0baaKqbGe aacaaIYaaabaGaaGOmaaaaaaqcfaOaeyypa0ZaaSaaaeaacaaIZaWa aiWaaqaabeqaamaabmaabaGaeqOSdiMaey4kaSIaaGOmaaGaayjkai aawMcaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSYa aeWaaeaacaaI0aGaeqOSdi2aaWbaaeqajuaibaGaaGOmaaaajuaGcq GHRaWkcaaIXaGaaGOnaiabek7aIjabgUcaRiaaigdacaaI2aaacaGL OaGaayzkaaGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHRa WkdaqadaqaaiaaiAdacqaHYoGydaahaaqabKqbGeaacaaIZaaaaKqb akabgUcaRiaaiodacaaI0aGaeqOSdi2aaWbaaeqajuaibaGaaGOmaa aajuaGcqGHRaWkcaaI1aGaaGioaiabek7aIjabgUcaRiaaiodacaaI 2aaacaGLOaGaayzkaaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaaK qbagaacqGHRaWkdaqadaqaaiaaisdacqaHYoGydaahaaqabKqbGeaa caaI0aaaaKqbakabgUcaRiaaikdacaaI4aGaeqOSdi2aaWbaaeqaju aibaGaaG4maaaajuaGcqGHRaWkcaaI2aGaaGioaiabek7aInaaCaaa beqcfasaaiaaikdaaaqcfaOaey4kaSIaaG4naiaaikdacqaHYoGycq GHRaWkcaaIZaGaaGOmaaGaayjkaiaawMcaaiabeI7aXbqaaiabgUca RmaabmaabaGaeqOSdi2aaWbaaeqajuaibaGaaGynaaaajuaGcqGHRa WkcaaI4aGaeqOSdi2aaWbaaeqajuaibaGaaGinaaaajuaGcqGHRaWk caaIYaGaaGynaiabek7aInaaCaaabeqcfasaaiaaiodaaaqcfaOaey 4kaSIaaG4maiaaiIdacqaHYoGydaahaaqabKqbGeaacaaIYaaaaKqb akabgUcaRiaaikdacaaI5aGaeqOSdiMaey4kaSIaaGymaiaaicdaai aawIcacaGLPaaaaaGaay5Eaiaaw2haaaqaaiabek7aInaacmaabaGa eqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkdaqadaqaai aaikdacqaHYoGycqGHRaWkcaaI0aaacaGLOaGaayzkaaGaeqiUdeNa ey4kaSYaaeWaaeaacqaHYoGydaahaaqabKqbGeaacaaIYaaaaKqbak abgUcaRiaaiodacqaHYoGycqGHRaWkcaaIZaaacaGLOaGaayzkaaaa caGL7bGaayzFaaWaaWbaaeqajuaibaGaaGOmaaaaaaaaaa@C4B4@

γ= σ 2 μ 1 = θ 2 +( 2β+4 )θ+( β 2 +3β+3 ) θ( θ+β+1 )( θ+β+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiqbeY7aTzaafaWaaSbaaKqbGeaacaaIXaaabeaaaaqcfaOaey ypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakab gUcaRmaabmaabaGaaGOmaiabek7aIjabgUcaRiaaisdaaiaawIcaca GLPaaacqaH4oqCcqGHRaWkdaqadaqaaiabek7aInaaCaaabeqcfasa aiaaikdaaaqcfaOaey4kaSIaaG4maiabek7aIjabgUcaRiaaiodaai aawIcacaGLPaaaaeaacqaH4oqCdaqadaqaaiabeI7aXjabgUcaRiab ek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaadaqadaqaaiabeI7aXj abgUcaRiabek7aIjabgUcaRiaaikdaaiaawIcacaGLPaaaaaaaaa@6776@

Behaviors of coefficient of variation (C.V), coefficient of skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@ , coefficient of kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B47@  and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@  of WGD have been prepared for different values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are presented in Tables 1, 2, 3, 4.

It is obvious from Table 1 that for a given β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ , the C.V. increases as the θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  increases, whereas for a given θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ , the C.V. decreases as the value of β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  increases. It is obvious from Table 2 that for a given θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ ( β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ ), ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@  decreases (increases) as the β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ ( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ ) increases. It is obvious from Table 3 that for a given θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ ( β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ ), the coefficient of Kurtosis ( β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@ ) decreases (increases) as the β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ ( θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ ) increases.

It is obvious from Table 4 that for a given β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ , the index of dispersion decreases as θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  increases. Similarly, for a given β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@ , the index of dispersion decreases as β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  increases.

      
      θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@
β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@

0.5

1

2

3

4

5

0.5

1.29099

1.32480

1.36083

1.37870

1.38888

1.39523

1

0.91473

0.93541

0.95917

0.97183

0.97938

0.98425

2

0.65263

0.66332

0.67700

0.68512

0.69034

0.69389

3

0.53783

0.54433

0.55328

0.55902

0.56291

0.56569

4

0.46948

0.47380

0.48007

0.48432

0.48734

0.48956

5

0.42269

0.42573

0.43033

0.43359

0.43598

0.43780

Table 1 Behavior of CV of WGD for varying values of parameters and

      θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@
β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@

0.5

1

2

3

4

5

0.5

2.375430

2.477646

2.599725

2.667337

2.708762

2.736002

1

1.698866

1.756288

1.831301

1.876396

1.905555

1.925486

2

1.236173

1.260866

1.298056

1.323613

1.341710

1.354931

3

1.033503

1.046136

1.067222

1.083251

1.095447

1.104854

4

0.911052

0.918239

0.931182

0.941812

0.950380

0.957291

5

0.825794

0.830208

0.838630

0.845981

0.852189

0.857388

Table 2 Behavior of ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqabaaajuaGbeaa aiaawIcacaGLPaaaaaa@3B56@ of WGD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@

   θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@

0.5

1

2

3

4

5

0.5

11.08000

11.84586

12.8368

13.42662

13.80492

14.06174

1

7.172516

7.469388

7.888469

8.159170

8.342689

8.472425

2

5.243856

5.330579

5.471074

5.574669

5.651707

5.710059

3

4.582041

4.617188

4.680000

4.731111

4.771968

4.804688

4

4.235147

4.252066

4.284545

4.313019

4.337119

4.357313

5

4.017509

4.026635

4.045120

4.062291

4.077505

4.090737

Table 3 Behavior of Kurtosis ( β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaaaa@387F@ ) of WGD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@

         θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@

0.5

1

2

3

4

5

0.5

2.500000

1.228571

0.595238

0.387205

0.284965

0.224615

1

2.342857

1.166667

0.575000

0.377778

0.279762

0.221429

2

2.190476

1.100000

0.550000

0.365079

0.272321

0.216667

3

2.121212

1.066667

0.535714

0.357143

0.267361

0.213333

4

2.083916

1.047619

0.526786

0.351852

0.263889

0.210909

5

2.061538

1.035714

0.520833

0.348148

0.261364

0.209091

Table 4 Behavior of γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379D@ of WGD for varying values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and

Reliability properties

In this section three important reliability properties namely hazard rate function, mean residual life function and stochastic ordering of WGD has been discussed

a. Hazard rate function

The survival (reliability) function of WGD can be expressed as

S( x;θ,β )=1 F 3 ( x;θ,β )= ( θx ) β e θx +( θ+β+1 )Γ( β,θx ) ( θ+β+1 )Γ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4uam aabmaabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGa ayzkaaGaeyypa0JaaGymaiabgkHiTiaadAeadaWgaaqcfasaaiaaio daaKqbagqaamaabmaabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOS digacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaqadaqaaiabeI7aXj aadIhaaiaawIcacaGLPaaadaahaaqabeaacqaHYoGyaaGaamyzamaa CaaabeqcfasaaiabgkHiTiabeI7aXjaadIhaaaqcfaOaey4kaSYaae WaaeaacqaH4oqCcqGHRaWkcqaHYoGycqGHRaWkcaaIXaaacaGLOaGa ayzkaaGaeu4KdC0aaeWaaeaacqaHYoGycaGGSaGaeqiUdeNaamiEaa GaayjkaiaawMcaaaqaamaabmaabaGaeqiUdeNaey4kaSIaeqOSdiMa ey4kaSIaaGymaaGaayjkaiaawMcaaiabfo5ahnaabmaabaGaeqOSdi gacaGLOaGaayzkaaaaaaaa@73AB@  (4.1.1)

The hazard (or failure) rate function, h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAai aacIcacaWG4bGaaiykaaaa@39C7@ of WGD is thus expressed as

h( x )= f 3 ( x;θ,β ) S( x;θ,β ) = x β1 θ β ( 1+θ+θx ) e θx ( θx ) β e θx +( θ+β+1 )Γ( β,θx ) ;x>0,θ>0,β>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiEaaGaayjkaiaawMcaaiabg2da9maalaaabaGaamOz amaaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oai abeI7aXjaacYcacqaHYoGyaiaawIcacaGLPaaaaeaacaWGtbWaaeWa aeaacaWG4bGaai4oaiabeI7aXjaacYcacqaHYoGyaiaawIcacaGLPa aaaaGaeyypa0ZaaSaaaeaacaWG4bWaaWbaaeqajuaibaGaeqOSdiMa eyOeI0IaaGymaaaajuaGcaaMc8UaeqiUde3aaWbaaeqajuaibaGaeq OSdigaaKqbaoaabmaabaGaaGymaiabgUcaRiabeI7aXjabgUcaRiab eI7aXjaadIhaaiaawIcacaGLPaaacaWGLbWaaWbaaeqajuaibaGaey OeI0IaeqiUdeNaamiEaaaaaKqbagaadaqadaqaaiabeI7aXjaadIha aiaawIcacaGLPaaadaahaaqabeaacqaHYoGyaaGaamyzamaaCaaabe qcfasaaiabgkHiTiabeI7aXjaadIhaaaqcfaOaey4kaSYaaeWaaeaa cqaH4oqCcqGHRaWkcqaHYoGycqGHRaWkcaaIXaaacaGLOaGaayzkaa GaaGPaVlabfo5ahnaabmaabaGaeqOSdiMaaiilaiabeI7aXjaadIha aiaawIcacaGLPaaaaaGaaGPaVlaaykW7caaMc8Uaai4oaiaaykW7ca aMc8UaamiEaiabg6da+iaaicdacaGGSaGaaGPaVlabeI7aXjabg6da +iaaicdacaGGSaGaaGPaVlabek7aIjabg6da+iaaicdaaaa@9A83@     (4.2)

The behavior of h(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAaiaacI cacaWG4bGaaiykaaaa@3939@  of WGD for different combinations of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are shown in Figure 3.

Figure 3 Behavior of the hazard function of WGD for different combinations θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycaGGUaaaaa@38D8@

Mean residual life function

The mean residual life function μ( x )=E( Xx| X>x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0Jaamyramaabmaa baGaamiwaiabgkHiTiaadIhadaabbaqaaiaadIfacqGH+aGpcaWG4b aacaGLhWoaaiaawIcacaGLPaaaaaa@4556@  of the WGD can be derived as

μ( x )= 1 S( x;θ,β ) x t f 3 ( t;θ,β )dtx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaeWaaeaacaWG4baacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacaaI XaaabaGaam4uamaabmaabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeq OSdigacaGLOaGaayzkaaaaamaapehabaGaamiDaiaaykW7caWGMbWa aSbaaeaacaaIZaaabeaadaqadaqaaiaadshacaGG7aGaeqiUdeNaai ilaiabek7aIbGaayjkaiaawMcaaiaadsgacaWG0bGaaGPaVlaaykW7 cqGHsislcaWG4baajuaibaGaamiEaaqaaiabg6HiLcqcfaOaey4kIi paaaa@5CA7@

= ( θx ) β ( θ+β+2 ) e θx +{ β 2 +βθ+2βθx( θ+β+1 ) }Γ( β,θx ) θ{ ( θx ) β e θx +( θ+β+1 )Γ( β,θx ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 ZaaSaaaeaadaqadaqaaiabeI7aXjaadIhaaiaawIcacaGLPaaadaah aaqabKqbGeaacqaHYoGyaaqcfa4aaeWaaeaacqaH4oqCcqGHRaWkcq aHYoGycqGHRaWkcaaIYaaacaGLOaGaayzkaaGaamyzamaaCaaabeqc fasaaiabgkHiTiabeI7aXjaadIhaaaqcfaOaey4kaSYaaiWaaeaacq aHYoGydaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiabek7aIjab eI7aXjabgUcaRiaaikdacqaHYoGycqGHsislcqaH4oqCcaWG4bWaae WaaeaacqaH4oqCcqGHRaWkcqaHYoGycqGHRaWkcaaIXaaacaGLOaGa ayzkaaaacaGL7bGaayzFaaGaeu4KdC0aaeWaaeaacqaHYoGycaGGSa GaeqiUdeNaamiEaaGaayjkaiaawMcaaaqaaiabeI7aXnaacmaabaWa aeWaaeaacqaH4oqCcaWG4baacaGLOaGaayzkaaWaaWbaaeqajuaiba GaeqOSdigaaKqbakaadwgadaahaaqabKqbGeaacqGHsislcqaH4oqC caWG4baaaKqbakabgUcaRmaabmaabaGaeqiUdeNaey4kaSIaeqOSdi Maey4kaSIaaGymaaGaayjkaiaawMcaaiabfo5ahnaabmaabaGaeqOS diMaaiilaiabeI7aXjaadIhaaiaawIcacaGLPaaaaiaawUhacaGL9b aaaaaaaa@8C17@

Clearly μ( 0 )= β( θ+β+2 ) θ( θ+β+1 ) = μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaeWaaeaacaaIWaaacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacqaH YoGydaqadaqaaiabeI7aXjabgUcaRiabek7aIjabgUcaRiaaikdaai aawIcacaGLPaaaaeaacqaH4oqCdaqadaqaaiabeI7aXjabgUcaRiab ek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaaaaGaeyypa0JafqiVd0 MbauaadaWgaaqcfasaaiaaigdaaeqaaaaa@517B@ . The behavior of μ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 MaaiikaiaadIhacaGGPaaaaa@3A90@ of the WGD for different combinations of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are shown in Figure 4.

Figure 4 Behavior of μ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maai ikaiaadIhacaGGPaaaaa@3A02@ of the WGD for different cpmbinations of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ and β. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GycaGGUaaaaa@38D8@
Figure 5 Fitted pdf plots of considered distribution for the given dataset.

c. Stochastic ordering

A random variable X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwaa aa@3761@  is said to be smaller than a random variable Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywaa aa@3762@  in the

  1. Stochastic order ( X st Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGZbGaamiDaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E4B@  if F X (x) F Y (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamiwaaqcfayabaGaaiikaiaadIhacaGGPaGaeyyz ImRaamOramaaBaaajuaibaGaamywaaqcfayabaGaaiikaiaadIhaca GGPaGaaGPaVlaaykW7aaa@4517@  for all x
  2. Hazard rate order ( X hr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaeaajugWaiaadIgacaWGYbaajuaGbeaa caWGzbaacaGLOaGaayzkaaGaaGPaVlaaykW7aaa@4254@ if h X (x) h Y (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aaBaaajuaibaGaamiwaaqcfayabaGaaiikaiaadIhacaGGPaGaeyyz ImRaamiAamaaBaaajuaibaGaamywaaqcfayabaGaaiikaiaadIhaca GGPaaaaa@4245@  for all x
  3. Mean residual life order ( X mrl Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaeaajugWaiaad2gacaWGYbGaamiBaaqc fayabaGaamywaaGaayjkaiaawMcaaaaa@4034@  if m X (x) m Y (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aaBaaajuaibaGaamiwaaqcfayabaGaaiikaiaadIhacaGGPaGaeyiz ImQaamyBamaaBaaajuaibaGaamywaaqcfayabaGaaiikaiaadIhaca GGPaaaaa@423E@  for all x
  4. Likelihood ratio order ( X lr Y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGybGaeyizIm6aaSbaaKqbGeaacaWGSbGaamOCaaqcfayabaGa amywaaGaayjkaiaawMcaaaaa@3E42@ if f X (x) f Y (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGMbWaaSbaaKqbGeaacaWGybaajuaGbeaacaGGOaGaamiEaiaa cMcaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaGbeaacaGGOaGaam iEaiaacMcaaaaaaa@408B@  decreases in x.

The following important interpretations due to Shaked & Shanthikumar13 are well known for establishing stochastic ordering of distributions.

X lr YX hr YX mrl Y X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaajuaGca WGybGaeyizIm6aaSbaaKqbGeaacaWGSbGaamOCaaqcfayabaGaamyw aiabgkDiElaadIfacqGHKjYOdaWgaaqcfasaaiaadIgacaWGYbaaju aGbeaacaWGzbGaeyO0H4TaamiwaiabgsMiJoaaBaaajuaibaGaamyB aiaadkhacaWGSbaajuaGbeaacaWGzbaabaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabgoDiFdGcbaqc faOaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaadIfacqGHKjYOdaWgaaqcfasaaiaado hacaWG0baajuaGbeaacaWGzbaaaaa@9BEC@

The WGD is ordered with respect to the strongest ‘likelihood ratio’ ordering as established in the following theorem.

Theorem

Let X~WGD( θ 1 , β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai aac6hacaqGxbGaae4raiaabseacaGGOaGaeqiUde3aaSbaaKqbGeaa caaIXaaajuaGbeaacaGGSaGaeqOSdi2aaSbaaKqbGeaacaaIXaaabe aajuaGcaGGPaaaaa@435E@  and Y~WGD( θ 2 , β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamywai aac6hacaqGxbGaae4raiaabseacaGGOaGaeqiUde3aaSbaaKqbGeaa caaIYaaabeaajuaGcaGGSaGaeqOSdi2aaSbaaKqbGeaacaaIYaaaju aGbeaacaGGPaaaaa@4361@ . If θ 1 > θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGH+aGpcqaH4oqCdaWgaaqc fasaaiaaikdaaeqaaaaa@3D9B@  and β 1 = β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIXaaajuaGbeaacqGH9aqpcqaHYoGydaWgaaqc fasaaiaaikdaaKqbagqaaaaa@3DFD@ ( or β 1 < β 2 and θ 1 = θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaqGVbGaaeOCaiaaykW7cqaHYoGydaWgaaqcfasaaiaaigdaaKqb agqaaiabgYda8iabek7aInaaBaaajuaibaGaaGOmaaqcfayabaGaaG PaVlaaykW7caqGHbGaaeOBaiaabsgacaaMc8UaaGPaVlabeI7aXnaa BaaajuaibaGaaGymaaqcfayabaGaeyypa0JaeqiUde3aaSbaaKqbGe aacaaIYaaajuaGbeaaaiaawIcacaGLPaaaaaa@5381@ , then X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfaaaa@3CB9@  and thus X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiAaiaadkhaaKqbagqaaiaadMfaaaa@3CB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamyBaiaadkhacaWGSbaajuaGbeaacaWG zbaaaa@3DAB@  and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaam4CaiaadshaaKqbagqaaiaadMfaaaa@3CC2@ .

Proof

We have

ln f X ( x; θ 1 , β 1 ) f Y ( x; θ 2, β 2 ) =ln( θ 1 β 1 ( θ 2 + β 2 +1 )Γ( β 2 ) θ 2 β 2 ( θ 1 + β 1 +1 )Γ( β 1 ) ) x β 1 β 2 ( 1+ θ 1 + θ 1 x 1+ θ 2 + θ 2 x )e( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaaMc8+aaSaaaeaacaWGMbWaaSbaaKqbGeaacaWGybaajuaG beaadaqadaqaaiaadIhacaGG7aGaeqiUde3aaSbaaKqbGeaacaaIXa aajuaGbeaacaGGSaGaeqOSdi2aaSbaaKqbGeaacaaIXaaajuaGbeaa aiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaGbe aadaqadaqaaiaadIhacaGG7aGaeqiUde3aaSbaaKqbGeaacaaIYaGa aiilaaqcfayabaGaeqOSdi2aaSbaaKqbGeaacaaIYaaajuaGbeaaai aawIcacaGLPaaaaaGaeyypa0JaciiBaiaac6gadaqadaqaamaalaaa baGaeqiUde3aa0baaKqbGeaacaaIXaaabaGaeqOSdiwcfa4aaSbaaK qbGeaacaaIXaaabeaaaaqcfa4aaeWaaeaacqaH4oqCdaWgaaqcfasa aiaaikdaaKqbagqaaiabgUcaRiabek7aInaaBaaajuaibaGaaGOmaa qcfayabaGaey4kaSIaaGymaaGaayjkaiaawMcaaiabfo5ahnaabmaa baGaeqOSdi2aaSbaaeaacaaIYaaabeaaaiaawIcacaGLPaaaaeaacq aH4oqCdaqhaaqcfasaaiaaikdaaeaacqaHYoGyjuaGdaWgaaqcfasa aiaaikdaaeqaaaaajuaGdaqadaqaaiabeI7aXnaaBaaajuaibaGaaG ymaaqcfayabaGaey4kaSIaeqOSdi2aaSbaaKqbGeaacaaIXaaajuaG beaacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeu4KdC0aaeWaaeaacq aHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaGaayjkaiaawMcaaaaa caaMc8oacaGLOaGaayzkaaGaaGPaVlaaykW7caWG4bWaaWbaaKqbGe qabaGaeqOSdiwcfa4aaSbaaKqbGeaajuaGdaWgaaqcfasaaiaaigda aeqaaaqabaGaeyOeI0IaeqOSdiwcfa4aaSbaaKqbGeaacaaIYaaabe aaaaqcfaOaaGPaVpaabmaabaWaaSaaaeaacaaIXaGaey4kaSIaeqiU de3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcqaH4oqCdaWgaa qcfasaaiaaigdaaKqbagqaaiaadIhaaeaacaaIXaGaey4kaSIaeqiU de3aaSbaaKqbGeaacaaIYaaajuaGbeaacqGHRaWkcqaH4oqCdaWgaa qcfasaaiaaikdaaKqbagqaaiaadIhaaaaacaGLOaGaayzkaaGaamyz aiabgkHiTmaabmaabaGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbe aacqGHsislcqaH4oqCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjk aiaawMcaaiaadIhaaaa@B88C@

Now ln f X ( x; θ 1 , β 1 ) f Y ( x; θ 2, β 2 ) =ln( θ 1 β 1 ( θ 2 + β 2 +1 )Γ( β 2 ) θ 2 β 2 ( θ 1 + β 1 +1 )Γ( β 1 ) )+( β 1 β 2 )lnx+ln( 1+ θ 1 + θ 1 x 1+ θ 2 + θ 2 x )( θ 1 θ 2 )x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaaMc8+aaSaaaeaacaWGMbWaaSbaaKqbGeaacaWGybaajuaG beaadaqadaqaaiaadIhacaGG7aGaeqiUde3aaSbaaKqbGeaacaaIXa aajuaGbeaacaGGSaGaeqOSdi2aaSbaaKqbGeaacaaIXaaajuaGbeaa aiaawIcacaGLPaaaaeaacaWGMbWaaSbaaKqbGeaacaWGzbaajuaGbe aadaqadaqaaiaadIhacaGG7aGaeqiUde3aaSbaaKqbGeaacaaIYaGa aiilaaqcfayabaGaeqOSdi2aaSbaaKqbGeaacaaIYaaajuaGbeaaai aawIcacaGLPaaaaaGaeyypa0JaciiBaiaac6gadaqadaqaamaalaaa baGaeqiUde3aa0baaKqbGeaacaaIXaaabaGaeqOSdiwcfa4aaSbaaK qbGeaacaaIXaaabeaaaaqcfa4aaeWaaeaacqaH4oqCdaWgaaqcfasa aiaaikdaaKqbagqaaiabgUcaRiabek7aInaaBaaajuaibaGaaGOmaa qcfayabaGaey4kaSIaaGymaaGaayjkaiaawMcaaiabfo5ahnaabmaa baGaeqOSdi2aaSbaaeaacaaIYaaabeaaaiaawIcacaGLPaaaaeaacq aH4oqCdaqhaaqcfasaaiaaikdaaeaacqaHYoGyjuaGdaWgaaqcfasa aiaaikdaaeqaaaaajuaGdaqadaqaaiabeI7aXnaaBaaajuaibaGaaG ymaaqcfayabaGaey4kaSIaeqOSdi2aaSbaaKqbGeaacaaIXaaajuaG beaacqGHRaWkcaaIXaaacaGLOaGaayzkaaGaeu4KdC0aaeWaaeaacq aHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaGaayjkaiaawMcaaaaa caaMc8oacaGLOaGaayzkaaGaaGPaVlaaykW7cqGHRaWkdaqadaqaai abek7aInaaBaaajuaibaGaaGymaaqcfayabaGaeyOeI0IaeqOSdi2a aSbaaKqbGeaacaaIYaaajuaGbeaaaiaawIcacaGLPaaaciGGSbGaai OBaiaadIhacqGHRaWkciGGSbGaaiOBaiaaykW7daqadaqaamaalaaa baGaaGymaiabgUcaRiabeI7aXnaaBaaajuaibaGaaGymaaqcfayaba Gaey4kaSIaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaacaWG4baa baGaaGymaiabgUcaRiabeI7aXnaaBaaajuaibaGaaGOmaaqcfayaba Gaey4kaSIaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaacaWG4baa aaGaayjkaiaawMcaaiabgkHiTmaabmaabaGaeqiUde3aaSbaaKqbGe aacaaIXaaajuaGbeaacqGHsislcqaH4oqCdaWgaaqcfasaaiaaikda aKqbagqaaaGaayjkaiaawMcaaiaadIhaaaa@BCFC@

This gives

d dx ln( f X ( x; θ 1 , β 1 ) f Y ( x; θ 2 , β 2 ) )= β 1 β 2 x + θ 1 θ 2 ( 1+ θ 1 + θ 1 x )( 1+ θ 2 + θ 2 x ) ( θ 1 θ 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaGaciiBaiaac6gadaqadaqaamaa laaabaGaamOzamaaBaaabaGaamiwaaqabaWaaeWaaeaacaWG4bGaai 4oaiabeI7aXnaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiabek7a InaaBaaajuaibaGaaGymaaqcfayabaaacaGLOaGaayzkaaaabaGaam OzamaaBaaabaGaamywaaqabaWaaeWaaeaacaWG4bGaai4oaiabeI7a XnaaBaaajuaibaGaaGOmaaqcfayabaGaaiilaiabek7aInaaBaaaju aibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaaaaGaayjkaiaawMca aiabg2da9maalaaabaGaeqOSdi2aaSbaaKqbGeaacaaIXaaajuaGbe aacqGHsislcqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaqaaiaa dIhaaaGaey4kaSYaaSaaaeaacqaH4oqCdaWgaaqcfasaaiaaigdaae qaaKqbakabgkHiTiabeI7aXnaaBaaajuaibaGaaGOmaaqcfayabaaa baWaaeWaaeaacaaIXaGaey4kaSIaeqiUde3aaSbaaKqbGeaacaaIXa aabeaajuaGcqGHRaWkcqaH4oqCdaWgaaqcfasaaiaaigdaaeqaaiaa dIhaaKqbakaawIcacaGLPaaadaqadaqaaiaaigdacqGHRaWkcqaH4o qCdaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRiabeI7aXnaaBaaa juaibaGaaGOmaaqabaGaamiEaaqcfaOaayjkaiaawMcaaaaacqGHsi sldaqadaqaaiabeI7aXnaaBaaajuaibaGaaGymaaqcfayabaGaeyOe I0IaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaaaiaawIcacaGLPa aacaGGUaaaaa@8992@

Thus, for β 1 = β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIXaaabeaajuaGcqGH9aqpcqaHYoGydaWgaaqc fasaaiaaikdaaKqbagqaaaaa@3DFD@  and θ 1 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde 3aaSbaaKqbGeaacaaIXaaajuaGbeaacqGHLjYScqaH4oqCdaWgaaqc fasaaiaaikdaaKqbagqaaaaa@3EE7@ , or ( β 1 < β 2 and θ 1 θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaiabgYda8iabek7a InaaBaaajuaibaGaaGOmaaqcfayabaGaaGPaVlaaykW7caqGHbGaae OBaiaabsgacaaMc8UaaGPaVlabeI7aXnaaBaaajuaibaGaaGymaaqc fayabaGaeyyzImRaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaaai aawIcacaGLPaaaaaa@50CF@ , d dx ln( f x ( x; θ 1 , β 1 ) f y ( x; θ 2 , β 2 ) )<0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaWGKbaabaGaamizaiaadIhaaaGaciiBaiaac6gadaqadaqaamaa laaabaGaamOzamaaBaaajuaibaGaamiEaaqcfayabaWaaeWaaeaaca WG4bGaai4oaiabeI7aXnaaBaaajuaibaGaaGymaaqabaqcfaOaaiil aiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaacaGLOaGaayzkaa aabaGaamOzamaaBaaajuaibaGaamyEaaqcfayabaWaaeWaaeaacaWG 4bGaai4oaiabeI7aXnaaBaaajuaibaGaaGOmaaqcfayabaGaaiilai abek7aInaaBaaajuaibaGaaGOmaaqcfayabaaacaGLOaGaayzkaaaa aaGaayjkaiaawMcaaiabgYda8iaaicdaaaa@5923@ . This shows that X lr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiBaiaadkhaaKqbagqaaiaadMfaaaa@3CB9@ and thus X hr Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamiAaiaadkhaaKqbagqaaiaadMfaaaa@3CB5@ , X mrl Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaamyBaiaadkhacaWGSbaajuaGbeaacaWG zbaaaa@3DAB@  and X st Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiwai abgsMiJoaaBaaajuaibaGaam4CaiaadshaaKqbagqaaiaadMfaaaa@3CC2@ . This shows flexibility of WGD over Garima distribution.

Estimation of parameters

Consider ( x 1 , x 2 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai aadIhadaWgaaqcfasaaiaaigdaaKqbagqaaiaacYcacaWG4bWaaSba aKqbGeaacaaIYaaajuaGbeaacaGGSaGaaGPaVlaac6cacaaMc8Uaai OlaiaaykW7caGGUaGaaiilaiaadIhadaWgaaqcfasaaiaad6gaaKqb agqaaiaacMcaaaa@489C@  be a random sample of size n from WGD (2.2). The natural log likelihood function can be expressed as

lnL=n[ βlnθln( θ+β+1 )ln( Γ( β ) ) ]+( β1 ) i=1 n ln( x i )+ i=1 n ln( 1+θ+θ x i ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBamaadmaabaGaeqOSdiMaaGPaVlGa cYgacaGGUbGaeqiUdeNaeyOeI0IaciiBaiaac6gadaqadaqaaiabeI 7aXjabgUcaRiabek7aIjabgUcaRiaaigdaaiaawIcacaGLPaaacqGH sislciGGSbGaaiOBamaabmaabaGaeu4KdC0aaeWaaeaacqaHYoGyai aawIcacaGLPaaaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGHRaWk daqadaqaaiabek7aIjabgkHiTiaaigdaaiaawIcacaGLPaaadaaeWb qaaiGacYgacaGGUbWaaeWaaeaacaWG4bWaaSbaaKqbGeaacaWGPbaa beaaaKqbakaawIcacaGLPaaacqGHRaWkdaaeWbqaaiGacYgacaGGUb WaaeWaaeaacaaIXaGaey4kaSIaeqiUdeNaey4kaSIaeqiUdeNaamiE amaaBaaajuaibaGaamyAaaqabaaajuaGcaGLOaGaayzkaaaajuaiba GaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHris5aaqcfasa aiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoacqGHsi slcaWGUbGaaGPaVlabeI7aXjaaykW7ceWG4bGbaebaaaa@84C2@

The maximum likelihood estimates ( θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaecaae aacqaH4oqCaiaawkWaaaaa@38FC@ , β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi Mbambaaaa@383F@ ) of (θ,β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaaiikai abeI7aXjaacYcacqaHYoGycaGGPaaaaa@3BE4@  is the solution of the following log likelihood equations.

lnL θ = nβ θ n θ+β+1 + i=1 n 1+ x i 1+θ+θ x i n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITciGGSbGaaiOBaiaadYeaaeaacqGHciITcqaH4oqCaaGa eyypa0ZaaSaaaeaacaWGUbGaeqOSdigabaGaeqiUdehaaiabgkHiTm aalaaabaGaamOBaaqaaiabeI7aXjabgUcaRiabek7aIjabgUcaRiaa igdaaaGaey4kaSYaaabCaeaadaWcaaqaaiaaigdacqGHRaWkcaWG4b WaaSbaaKqbGeaacaWGPbaajuaGbeaaaeaacaaIXaGaey4kaSIaeqiU deNaey4kaSIaeqiUdeNaamiEamaaBaaajuaibaGaamyAaaqcfayaba aaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyye IuoacqGHsislcaWGUbGaaGPaVlaaykW7ceWG4bGbaebacqGH9aqpca aIWaaaaa@674E@

lnL β =nlnθ n θ+β+1 nψ( β )+ i=1 n ln( x i ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqGHciITciGGSbGaaiOBaiaadYeaaeaacqGHciITcqaHYoGyaaGa eyypa0JaamOBaiGacYgacaGGUbGaeqiUdeNaeyOeI0YaaSaaaeaaca WGUbaabaGaeqiUdeNaey4kaSIaeqOSdiMaey4kaSIaaGymaaaacqGH sislcaWGUbGaaGPaVlaaykW7cqaHipqEdaqadaqaaiabek7aIbGaay jkaiaawMcaaiabgUcaRmaaqahabaGaciiBaiaac6gadaqadaqaaiaa dIhadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaaqcfa saaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoacqGH 9aqpcaaIWaaaaa@645C@

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiEayaara aaaa@370B@  being the sample mean and ψ(β) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK Naaiikaiabek7aIjaacMcaaaa@3B4C@  is the digamma function defined as ψ( β )= d dβ lnΓ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiYdK 3aaeWaaeaacqaHYoGyaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaa dsgaaeaacaWGKbGaeqOSdigaaiGacYgacaGGUbGaeu4KdC0aaeWaae aacqaHYoGyaiaawIcacaGLPaaaaaa@467B@ . Since these two log likelihood equations are not in closed forms, they cannot be solved analytically. However, the MLE’s ( θ ^ , β ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacuaH4oqCgaqcaiaacYcacuaHYoGygaqcaaGaayjkaiaawMcaaaaa @3C34@  of ( θ,β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzkaaaaaa@3C14@  can be computed directly by solving the natural log likelihood equation using Newton–Raphson method available in R–software till sufficiently close estimates of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde Nbambaaaa@3855@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi Mbambaaaa@383F@ are obtained. In this paper starting values of parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are θ=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGimaiaac6cacaaI1aaaaa@3B6B@  and β=1.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi Maeyypa0JaaGymaiaac6cacaaI1aaaaa@3B57@ , respectively.

Data analysis

In this section a real lifetime data has been considered for the goodness of fit of WGD over one–parameter and two–parameter life time distributions. The data is regarding the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm, available in Bader & Priest.14

1.312     1.314      1.479      1.552      1.700      1.803      1.861      1.865      1.944      1.958      1.966      1.997      2.006      2.021      2.027      2.055      2.063      2.098      2.140      2.179      2.224      2.240      2.253      2.270      2.272      2.274      2.301      2.301      2.359      2.382      2.382      2.426      2.434      2.435      2.478      2.490      2.511      2.514      2.535      2.554      2.566      2.570      2.586      2.629      2.633      2.642      2.648      2.684      2.697      2.726      2.770      2.773      2.800      2.809      2.818      2.821      2.848      2.880      2.954      3.012      3.067      3.084      3.090      3.096      3.128      3.233      3.433      3.585      3.585

The goodness of fit of WGD has been compared with one parameter exponential, Lindley and Garima distributions and two–parameter Gompertz distribution, generalized exponential distribution (GED) introduced by Gupta & Kundu15, lognormal distribution and WLD. The pdf and cdf of Gompertz distributions, lognormal and GED are presented in Table 5. The ML estimates of parameters, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , Akaike Information criteria (AIC), K–S statistics and p–value of distributions for the considered dataset are presented in Table 6. The AIC and K–S Statistics are calculated using the formulae: AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaaikdacaWGRbaaaa@40D2@  and K-S= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaae4sai aab2cacaqGtbGaeyypa0ZaaCbeaeaacaqGtbGaaeyDaiaabchaaKqb GeaacaWG4baajuaGbeaadaabdaqaaiaadAeadaWgaaqcfasaaiaad6 gaaKqbagqaamaabmaabaGaamiEaaGaayjkaiaawMcaaiabgkHiTiaa dAeadaWgaaqcfasaaiaaicdaaKqbagqaamaabmaabaGaamiEaaGaay jkaiaawMcaaaGaay5bSlaawIa7aaaa@4C9E@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@  = the number of parameters, n = the sample size, F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamOBaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaaaaa@3BA5@ is the empirical (sample) cumulative distribution function, and F 0 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGimaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaaaaa@3B6C@  is the theoretical cumulative distribution function. The best distribution is the distribution corresponding to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC, and K–S statistics.

It is quite obvious from table 6 that WGD is competing well with two–parameter lifetime distributions and gives quite satisfactory fit.

Distributions

pdf

cdf

WLD

f( x;θ,β )= θ β+1 ( θ+β ) x β1 Γ( β ) ( 1+x ) e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWcbeqaaiabek7aIjabgU caRiaaigdaaaaakeaadaqadaqaaiabeI7aXjabgUcaRiabek7aIbGa ayjkaiaawMcaaaaacaaMc8+aaSaaaeaacaWG4bWaaWbaaSqabeaacq aHYoGycqGHsislcaaIXaaaaaGcbaGaeu4KdC0aaeWaaeaacqaHYoGy aiaawIcacaGLPaaaaaGaaGPaVpaabmaabaGaaGymaiabgUcaRiaadI haaiaawIcacaGLPaaacaaMc8UaamyzamaaCaaaleqabaGaeyOeI0Ia eqiUdeNaaGPaVlaadIhaaaGccaaMc8UaaGPaVdaa@6575@

F( x;θ,β )=1 ( θ+β )Γ( β,θx )+ ( θx ) β e θx ( θ+β )Γ( β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0JaaGymaiabgkHiTmaalaaabaWaaeWaaeaacqaH4oqCcq GHRaWkcqaHYoGyaiaawIcacaGLPaaacqqHtoWrdaqadaqaaiabek7a IjaacYcacqaH4oqCcaaMc8UaamiEaaGaayjkaiaawMcaaiabgUcaRm aabmaabaGaeqiUdeNaaGPaVlaadIhaaiaawIcacaGLPaaadaahaaWc beqaaiabek7aIbaakiaadwgadaahaaWcbeqaaiabgkHiTiabeI7aXj aaykW7caWG4baaaaGcbaWaaeWaaeaacqaH4oqCcqGHRaWkcqaHYoGy aiaawIcacaGLPaaacqqHtoWrdaqadaqaaiabek7aIbGaayjkaiaawM caaaaacaaMc8UaaGPaVdaa@6C44@

Lognormal

f( x;θ,β )= 1 2π βx e 1 2 ( logxθ β ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaGaeqiWda haleqaaOGaeqOSdiMaamiEaaaacaaMc8UaamyzamaaCaaaleqabaGa eyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaaba GaciiBaiaac+gacaGGNbGaamiEaiabgkHiTiabeI7aXbqaaiabek7a IbaaaiaawIcacaGLPaaadaahaaadbeqaaiaaikdaaaaaaaaa@54E4@

F( x;θ,β )=ϕ( logxθ β ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0Jaeqy1dy2aaeWaaeaadaWcaaqaaiGacYgacaGGVbGaai 4zaiaadIhacqGHsislcqaH4oqCaeaacqaHYoGyaaaacaGLOaGaayzk aaaaaa@4A85@

GED

f( x;θ,β )=θβ ( 1 e θx ) β1 e θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0JaeqiUdeNaaGPaVlabek7aInaabmaabaGaaGymaiabgk HiTiaadwgadaahaaWcbeqaaiabgkHiTiabeI7aXjaaykW7caWG4baa aaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaHYoGycqGHsislcaaIXa aaaOGaaGPaVlaadwgadaahaaWcbeqaaiabgkHiTiabeI7aXjaaykW7 caWG4baaaaaa@58DF@

F( x;θ,β )= ( 1 e θx ) β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0ZaaeWaaeaacaaIXaGaeyOeI0IaamyzamaaCaaaleqaba GaeyOeI0IaeqiUdeNaamiEaaaaaOGaayjkaiaawMcaamaaCaaaleqa baGaeqOSdigaaaaa@48D3@

Gompertz

f( x;θ,β )=θ e βx θ β ( e βx 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0JaeqiUdeNaaGPaVlaadwgadaahaaWcbeqaaiabek7aIj aaykW7caWG4bGaeyOeI0YaaSaaaeaacqaH4oqCaeaacqaHYoGyaaWa aeWaaeaacaWGLbWaaWbaaWqabeaacqaHYoGycaaMc8UaamiEaaaali abgkHiTiaaigdaaiaawIcacaGLPaaaaaaaaa@5485@

F( x;θ,β )=1 e θ β ( e βx 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiEaiaacUdacqaH4oqCcaGGSaGaeqOSdigacaGLOaGaayzk aaGaeyypa0JaaGymaiabgkHiTiaadwgadaahaaWcbeqaaiabgkHiTi aaykW7caaMc8+aaSaaaeaacqaH4oqCaeaacqaHYoGyaaWaaeWaaeaa caWGLbWaaWbaaWqabeaacqaHYoGycaaMc8UaamiEaaaaliabgkHiTi aaigdaaiaawIcacaGLPaaaaaGccaaMc8UaaGPaVdaa@54D9@

Table 5 The pdf and the cdf of fitted distributions

Distributions

ML Estimates

2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiGacYgacaGGUbGaamitaaaa@3A54@

AIC

K-S

 

P-value

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@

β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdigaaa@3797@

WGD

9.3798

22.3473

101.94

105.94

0.057

0.979

WLD

9.6265

22.8938

101.95

105.95

0.059

0.973

Lognormal

0.8751

0.2124

102.72

106.73

0.103

0.713

GED

2.0331

87.2847

109.24

113.24

0.087

0.613

Gompertz

0.0080

2.0420

107.25

111.250

0.085

0.673

Garima

0.5863

-------

251.33

253.33

0.4381

0.000

Lindley

0.0702

-------

238.38

240.38

0.401

0.000

Exponential

0.4079

-------

261.73

263.73

0.448

0.000

Table 6 MLE’s, - 2ln L, AIC, K-S Statistics and p-values of the fitted distributions

Conclusion

A two–parameter weighted Garima distribution(WGD) which includes one parameter Garima distribution proposed by Shanker5 has been suggested for modeling lifetime data from engineering. Its statistical properties including shapes of probability density function for different combinations of parameters, coefficients of variation, skweness, kurtosis, and index of dispersion have been explained. Reliability measures including hazard rate function, mean residual life function and the stochastic ordering have been studied. Estimation of parameters has been discussed using maximum likelihood. The goodness of fit of WGD has been explained with a real lifetime data and the fit has been found to be quite satisfactory over exponential, Lindley and Garima, Gompertz, generalized exponential, lognormal and weighted Lindley distributions.

Acknowledgements

Authors are grateful to the editor–in–chief of the journal and the anonymous reviewer for fruitful comments on the paper.

Conflict of interest

Author declares there is no conflict of interest towards this manuscript.

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