ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 7 - 2016
On Modeling Of Lifetime Data Using Three-Parameter Generalized Lindley and Generalized Gamma Distributions
Rama Shanker* and Kamlesh Kumar Shukla
Rama Shanker* and Kamlesh Kumar Shukla
Received: September 19, 2016 | Published: December 09, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Eritrea, Email:
Citation: Shanker R, Shukla KK (2016) On Modeling Of Lifetime Data Using Three-Parameter Generalized Lindley and Generalized Gamma Distributions. Biom Biostat Int J 4(7): 00117. DOI: 10.15406/bbij.2016.04.00117

Abstract

The analysis and modeling of lifetime data are crucial in almost all applied sciences including behavioral sciences, medicine, insurance, engineering, and finance, amongst others. In this paper an attempt has been made for comparative study of generalized Lindley distribution (GLD) introduced by Zakerzadeh & Dolati [1] and generalized gamma distribution (GGD) introduced by Stacy [2] for modeling lifetime data from different fields of knowledge. The goodness of fit for both GLD and GGD , based on maximum likelihood estimates, shows that GGD gives much closer fit than GLD in majority of data sets and hence GGD can be considered as an important tool for modeling lifetime data over GLD.

Keywords: Generalized Lindley distribution; Generalized Gamma distribution; Lifetime data, Estimation of parameter; Goodness of fit

Introduction

In reliability analysis the time to the occurrence of event of interest is the lifetime or survival time or failure time. The event may be failure of a piece of equipment, death of a person, development (or remission) of symptoms of disease, health code violation (or compliance). The statistical analysis and the modeling of lifetime data are crucial for statisticians and research workers in almost all applied sciences including behavioral sciences, engineering, medical science/biological science, insurance and finance, amongst others.

A number of lifetime distributions are available for modeling lifetime data in statistics including exponential distribution, gamma distribution, Lindley distribution, Weibull distribution and their generalizations, some amongst others.

In this paper various lifetime data have been considered for modeling using three- parameter generalized Lindley distribution (GLD) and generalized gamma distribution (GGD) because gamma, Lindley and exponential distributions are particular cases of GLD whereas gamma, Weibull and exponential distributions are particular cases of GGD.

Generalized Lindley Distribution

The probability density function of three-parameter generalized Lindley distribution (GLD) introduced by Zakerzadeh & Dolati [1] having parameters α,β,andθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde MaaGPaVlaacYcacqaHYoGycaaMc8UaaiilaiaaykW7caqGHbGaaeOB aiaabsgacaaMc8UaaGPaVlabeI7aXbaa@474D@  is given by

f 1 ( x;α,β,θ )= θ α+1 ( β+θ ) x α1 Γ( α+1 ) ( α+βx ) e θx ;x>0,α>0,β>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaKqbao aaBaaabaqcLbmacaaIXaaajuaGbeaadaqadaqaaiaadIhacaGG7aGa eqySdeMaaiilaiabek7aIjaacYcacqaH4oqCaiaawIcacaGLPaaacq GH9aqpdaWcaaqaaiabeI7aXnaaCaaabeqaaKqzadGaeqySdeMaey4k aSIaaGymaaaaaKqbagaadaqadaqaaiabek7aIjabgUcaRiabeI7aXb GaayjkaiaawMcaaaaadaWcaaqaaiaadIhalmaaCaaajuaGbeqaaKqz adGaeqySdeMaeyOeI0IaaGymaaaaaKqbagaacqqHtoWrdaqadaqaai abeg7aHjabgUcaRiaaigdaaiaawIcacaGLPaaaaaWaaeWaaeaacqaH XoqycqGHRaWkcqaHYoGycaaMc8UaamiEaaGaayjkaiaawMcaaiaayk W7caWGLbWaaWbaaeqabaqcLbmacqGHsislcqaH4oqCcaaMc8UaamiE aaaajuaGcaGG7aGaaGPaVlaaykW7caWG4bGaeyOpa4JaaGimaiaacY cacqaHXoqycqGH+aGpcaaIWaGaaiilaiabek7aIjabg6da+iaaicda caGGSaGaeqiUdeNaeyOpa4JaaGimaaaa@82D3@ (2.1)

Clearly the gamma distribution, the Lindley [3] distribution and the exponential distribution are particular cases of (2.1) for ( β=0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGycqGH9aqpcaaIWaaacaGLOaGaayzkaaaaaa@3B6E@ , ( α=β=1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHXoqycqGH9aqpcqaHYoGycqGH9aqpcaaIXaaacaGLOaGaayzk aaaaaa@3E14@  and ( α=1,β=0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHXoqycqGH9aqpcaaIXaGaaiilaiabek7aIjabg2da9iaaicda aiaawIcacaGLPaaaaaa@3F7E@  respectively. The discussion about its properties, estimation of parameters and applications are available in Zakerzadeh & Dolati [1]. Ghitany et al. [4] have detailed study about various properties of Lindley distribution, estimation of parameter and application for modeling waiting time data in a bank. Shanker et al. [5] have detailed and comparative study about modeling of lifetime data using one parameter Lindley and exponential distributions.

 The corresponding distribution function of the GLD can be     obtained as

F 1 ( x;α,β,θ )=1 α( β+θ )Γ( α,θx )+β ( θx ) α e θx ( β+θ )Γ( α+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eg7aHjaacYcacqaHYoGycaGGSaGaeqiUdehacaGLOaGaayzkaaGaey ypa0JaaGymaiabgkHiTiaaykW7daWcaaqaaiabeg7aHnaabmaabaGa eqOSdiMaey4kaSIaeqiUdehacaGLOaGaayzkaaGaeu4KdC0aaeWaae aacqaHXoqycaGGSaGaeqiUdeNaamiEaaGaayjkaiaawMcaaiabgUca Riabek7aInaabmaabaGaeqiUdeNaamiEaaGaayjkaiaawMcaamaaCa aabeqcfasaaiabeg7aHbaajuaGcaWGLbWaaWbaaeqajuaibaGaeyOe I0IaeqiUdeNaaGPaVlaadIhaaaaajuaGbaWaaeWaaeaacqaHYoGycq GHRaWkcqaH4oqCaiaawIcacaGLPaaacqqHtoWrdaqadaqaaiabeg7a HjabgUcaRiaaigdaaiaawIcacaGLPaaaaaaaaa@723D@ ; x>0,α>0,β>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abg6da+iaaicdacaGGSaGaaGPaVlabeg7aHjabg6da+iaaicdacaGG SaGaeqOSdiMaeyOpa4JaaGimaiaacYcacqaH4oqCcqGH+aGpcaaIWa aaaa@471A@ (2.2)

Where Γ( α,z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqqHto WrjuaGdaqadaqaaKqzGeGaeqySdeMaaiilaiaadQhaaKqbakaawIca caGLPaaaaaa@3E6F@  is the upper incomplete gamma function defined as

( α,z )= z e y y α1 dy;α>0,z0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHXoqycaGGSaGaamOEaaGaayjkaiaawMcaaiabg2da9maapeha baGaamyzamaaCaaabeqaaKqzadGaeyOeI0IaamyEaaaajuaGcaaMc8 UaamyEamaaCaaabeqaaKqzadGaeqySdeMaeyOeI0IaaGymaaaajuaG caaMc8UaamizaiaadMhacaaMc8Uaai4oaiabeg7aHjabg6da+iaaic dacaGGSaGaaGPaVlaaykW7caWG6bGaeyyzImRaaGimaaqaaKqzadGa amOEaaqcfayaaKqzadGaeyOhIukajuaGcqGHRiI8aaaa@615D@ (2.3)

Recently Shanker [6] has detailed study about GLD and obtained expressions for coefficient of variation, skewness, kurtosis and index of dispersion. Shanker [6] has also studied its hazard rate function and the mean residual life function.

Generalized Gamma Distribution

The probability density function of three-parameter generalized gamma distribution (GGD) introduced by Stacy [2] having parameters α,β,andθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde MaaGPaVlaacYcacqaHYoGycaaMc8UaaiilaiaaykW7caqGHbGaaeOB aiaabsgacaaMc8UaaGPaVlabeI7aXbaa@474D@  is given by

f 2 ( x;α,β,θ )= β θ α Γ( α ) x βα1 e θ x β ;x>0,α>0,β>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaabaqcLbmacaaIYaaajuaGbeaadaqadaqaaiaadIhacaGG7aGa eqySdeMaaiilaiabek7aIjaacYcacqaH4oqCaiaawIcacaGLPaaacq GH9aqpdaWcaaqaaiabek7aIjaaykW7cqaH4oqCdaahaaqabeaajugW aiabeg7aHbaaaKqbagaacqqHtoWrdaqadaqaaKqzadGaeqySdegaju aGcaGLOaGaayzkaaaaaiaadIhalmaaCaaajuaGbeqaaKqzadGaeqOS diMaaGPaVlabeg7aHjabgkHiTiaaigdaaaGaaGPaVNqbakaadwgada ahaaqabeaajugWaiabgkHiTiabeI7aXjaaykW7caWG4bWcdaahaaqc fayabeaajugWaiabek7aIbaaaaqcfaOaai4oaiaaykW7caaMc8Uaam iEaiabg6da+iaaicdacaGGSaGaeqySdeMaeyOpa4JaaGimaiaacYca cqaHYoGycqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6da+iaaicdaaa a@7D33@ (3.1)

 Where α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi gaaa@3825@  are the shape parameter and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the scale parameter. Clearly the gamma distribution, the Weibull distribution and the exponential distribution are particular cases of (3.1) for ( β=1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGycqGH9aqpcaaIXaaacaGLOaGaayzkaaaaaa@3B6F@ , ( α=1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHXoqycqGH9aqpcaaIXaaacaGLOaGaayzkaaaaaa@3B6D@  and ( α=β=1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHXoqycqGH9aqpcqaHYoGycqGH9aqpcaaIXaaacaGLOaGaayzk aaaaaa@3E14@  respectively. Detailed discussion about GGD is available in Stacy [2] and parametric estimation for the GGD is available in Stacy & Mihram [7].

 The cumulative distribution function of the GGD is thus given by

F 2 ( x;α,β,θ )=1 Γ( α,θ x β ) Γ( α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eg7aHjaacYcacqaHYoGycaGGSaGaeqiUdehacaGLOaGaayzkaaGaey ypa0JaaGymaiabgkHiTmaalaaabaGaeu4KdC0aaeWaaeaacqaHXoqy caGGSaGaeqiUdeNaaGPaVlaadIhadaahaaqabKqbGeaacqaHYoGyaa aajuaGcaGLOaGaayzkaaaabaGaeu4KdC0aaeWaaeaacqaHXoqyaiaa wIcacaGLPaaaaaaaaa@55CE@ (3.2)

Where Γ( α,z ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeu4KdC 0aaeWaaeaacqaHXoqycaGGSaGaamOEaaGaayjkaiaawMcaaaaa@3CC3@ is the upper incomplete gamma function defined in (2.3)

Maximum Likelihood Estimation

Maximum likelihood estimates of the parameters of GLD

Assuming ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaeaajugWaiaaigdaaKqbagqaaiaacYcacaaMc8Ua amiEamaaBaaabaqcLbmacaaIYaaajuaGbeaacaGGSaGaaGPaVlaadI halmaaBaaajuaGbaqcLbmacaaIZaaajuaGbeaacaGGSaGaaGPaVlaa ykW7caGGUaGaaiOlaiaac6cacaaMc8UaaGPaVlaacYcacaWG4bWaaS baaeaajugWaiaad6gaaKqbagqaaaGaayjkaiaawMcaaaaa@554D@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  from GLD (2.1), the likelihood function, L of GLD is given by

L= ( θ α+1 β+θ ) n 1 ( Γ( α+1 ) ) n i=1 n x i α1 ( α+β x i ) e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabeaajugWaiab eg7aHjabgUcaRiaaigdaaaaajuaGbaGaeqOSdiMaey4kaSIaeqiUde haaaGaayjkaiaawMcaamaaCaaabeqaaKqzadGaamOBaaaajuaGdaWc aaqaaiaaigdaaeaadaqadaqaaiabfo5ahnaabmaabaGaeqySdeMaey 4kaSIaaGymaaGaayjkaiaawMcaaaGaayjkaiaawMcaamaaCaaabeqa aKqzadGaamOBaaaaaaqcfa4aaebCaeaacaWG4bWaaSbaaeaajugWai aadMgaaKqbagqaamaaCaaabeqaaKqzadGaeqySdeMaeyOeI0IaaGym aaaajuaGdaqadaqaaiabeg7aHjabgUcaRiabek7aIjaadIhalmaaBa aajuaGbaqcLbmacaWGPbaajuaGbeaaaiaawIcacaGLPaaaaeaajugW aiaadMgacqGH9aqpcaaIXaaajuaGbaqcLbmacaWGUbaajuaGcqGHpi s1aiaaykW7caWGLbWcdaahaaqcfayabeaajugWaiabgkHiTiaad6ga caaMc8UaeqiUdeNaaGPaVlqadIhagaqeaaaaaaa@7B6B@ ; x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiEay aaraaaaa@3799@  being the sample mean

The natural log likelihood function is thus obtained as

lnL=n[ ( α+1 )lnθln( β+θ )ln( Γ( α+1 ) ) ]+( α1 ) i=1 n ln( x i ) + i=1 n ln( α+β x i ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBamaadmaabaWaaeWaaeaacqaHXoqy cqGHRaWkcaaIXaaacaGLOaGaayzkaaGaciiBaiaac6gacqaH4oqCcq GHsislciGGSbGaaiOBamaabmaabaGaeqOSdiMaey4kaSIaeqiUdeha caGLOaGaayzkaaGaeyOeI0IaciiBaiaac6gadaqadaqaaiabfo5ahn aabmaabaGaeqySdeMaey4kaSIaaGymaaGaayjkaiaawMcaaaGaayjk aiaawMcaaaGaay5waiaaw2faaiabgUcaRmaabmaabaGaeqySdeMaey OeI0IaaGymaaGaayjkaiaawMcaamaaqahabaGaciiBaiaac6gadaqa daqaaiaadIhadaWgaaqaaKqzadGaamyAaaqcfayabaaacaGLOaGaay zkaaaabaqcLbmacaWGPbGaeyypa0JaaGymaaqcfayaaKqzadGaamOB aaqcfaOaeyyeIuoacqGHRaWkdaaeWbqaaiGacYgacaGGUbWaaeWaae aacqaHXoqycqGHRaWkcqaHYoGycaWG4bWcdaWgaaqcfayaaKqzadGa amyAaaqcfayabaaacaGLOaGaayzkaaaabaqcLbmacaWGPbGaeyypa0 JaaGymaaqcfayaaKqzadGaamOBaaqcfaOaeyyeIuoacqGHsislcaWG UbGaaGPaVlabeI7aXjaaykW7ceWG4bGbaebaaaa@8C9F@

The maximum likelihood estimate (MLE) θ ^ , α ^ , β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacaaMc8UaaGPaVlaacYcacuaHXoqygaqcaiaaykW7caaMc8Ua aiilaiqbek7aIzaajaaaaa@4336@  of parameters θ,α,β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7cqaHXoqycaaMc8UaaGPa VlaacYcacqaHYoGyaaa@461C@  of GLD can be obtained by solving the natural log likelihood equation using R software (Package Stat 4).

Maximum likelihood estimates of the parameters of GGD

Let ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaGaaiilaiaa ykW7caWG4bWcdaWgaaqcfayaaKqzadGaaGOmaaqcfayabaGaaiilai aaykW7caWG4bWaaSbaaeaajugWaiaaiodaaKqbagqaaiaacYcacaaM c8UaaGPaVlaac6cacaGGUaGaaiOlaiaaykW7caaMc8UaaiilaiaadI hadaWgaaqaaKqzadGaamOBaaqcfayabaaacaGLOaGaayzkaaaaaa@55E6@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  from GGD (3.1). Then

f( x i )= β θ α Γ( α ) ( x i ) βα1 e θ ( x i ) β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiEaSWaaSbaaKqbagaajugWaiaadMgaaKqbagqaaaGa ayjkaiaawMcaaiabg2da9maalaaabaGaeqOSdiMaaGPaVlabeI7aXn aaCaaabeqaaKqzadGaeqySdegaaaqcfayaaiabfo5ahnaabmaabaqc LbmacqaHXoqyaKqbakaawIcacaGLPaaaaaWaaeWaaeaacaWG4bWcda WgaaqcfayaaKqzadGaamyAaaqcfayabaaacaGLOaGaayzkaaWaaWba aeqabaqcLbmacqaHYoGycaaMc8UaeqySdeMaeyOeI0IaaGymaaaaju aGcaaMc8UaamyzamaaCaaabeqaaKqzadGaeyOeI0IaeqiUdeNaaGPa VVWaaeWaaKqbagaajugWaiaadIhalmaaBaaajuaGbaqcLbmacaWGPb aajuaGbeaaaiaawIcacaGLPaaadaahaaqabeaajugWaiabek7aIbaa aaaaaa@6DEA@

This gives

lnf( x i )=lnβ+αlnθln( Γ( α ) )+( βα1 )ln( x i )θ ( x i ) β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGMbWaaeWaaeaacaWG4bWcdaWgaaqcfayaaKqzadGaamyA aaqcfayabaaacaGLOaGaayzkaaGaeyypa0JaciiBaiaac6gacqaHYo GycqGHRaWkcqaHXoqyciGGSbGaaiOBaiabeI7aXjabgkHiTiGacYga caGGUbWaaeWaaeaacqqHtoWrdaqadaqaaKqzadGaeqySdegajuaGca GLOaGaayzkaaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacqaHYoGy cqaHXoqycqGHsislcaaIXaaacaGLOaGaayzkaaGaciiBaiaac6gada qadaqaaiaadIhadaWgaaqaaiaadMgaaeqaaaGaayjkaiaawMcaaiab gkHiTiabeI7aXnaabmaabaGaamiEaSWaaSbaaKqbagaajugWaiaadM gaaKqbagqaaaGaayjkaiaawMcaamaaCaaabeqaaKqzadGaeqOSdiga aaaa@6CD3@

Thus the natural log likelihood function of the GGD is given by

lnL=n[ lnβ+αlnθln( Γ( α ) ) ]+( βα1 ) i=1 n ln( x i ) θ i=1 n x i β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBamaadmaabaGaciiBaiaac6gacqaH YoGycqGHRaWkcqaHXoqyciGGSbGaaiOBaiabeI7aXjabgkHiTiGacY gacaGGUbWaaeWaaeaacqqHtoWrdaqadaqaaKqzadGaeqySdegajuaG caGLOaGaayzkaaaacaGLOaGaayzkaaaacaGLBbGaayzxaaGaey4kaS YaaeWaaeaacqaHYoGycaaMc8UaeqySdeMaeyOeI0IaaGymaaGaayjk aiaawMcaamaaqahabaGaciiBaiaac6gadaqadaqaaiaadIhadaWgaa qaaiaadMgaaeqaaaGaayjkaiaawMcaaaqaaKqzadGaamyAaiabg2da 9iaaigdaaKqbagaajugWaiaad6gaaKqbakabggHiLdGaeyOeI0Iaeq iUde3aaabCaeaacaWG4bWcdaWgaaqcfayaaKqzadGaamyAaaqcfaya baWcdaahaaqcfayabeaajugWaiabek7aIbaaaKqbagaajugWaiaadM gacqGH9aqpcaaIXaaajuaGbaqcLbmacaWGUbaajuaGcqGHris5aaaa @7D65@

The maximum likelihood estimate (MLE) θ ^ , α ^ , β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacaaMc8UaaGPaVlaacYcacuaHXoqygaqcaiaaykW7caaMc8Ua aiilaiqbek7aIzaajaaaaa@4336@  of parameters θ,α,β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaaGPaVlaaykW7caGGSaGaaGPaVlaaykW7cqaHXoqycaaMc8UaaGPa VlaacYcacqaHYoGyaaa@461C@  of GGD can be obtained by solving the natural log likelihood equation using R software (Package Stat 4).

Goodness of Fit and Applications

In this section, the goodness of fit and applications of GLD and GGD have been discussed for several lifetime data In order to compare GLD and GGD, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@  and K-S Statistics Kolmogorov-Smirnov Statistics) for eighteen data sets have been computed and presented in Table 1.

Model

ML Estimates

-2ln L

K-S
Statistic

P-Value

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqySde MbaKaaaaa@3833@

β ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqOSdi MbaKaaaaa@3835@

θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@

Data 1

GLD

17. 1792

14. 3378

11.7653

47.784

0.809

0

GGD

0.6831

7.2644

0.0176

29.238

0.796

0

Data 2

GLD

7.0755

0.4492

0.1152

914.95

0.098

0.281

GGD

4.8293

1.3071

0.0188

912.437

0.087

0.429

Data 3

GLD

3.0404

3.5804

0.0557

226.06

0.123

0.833

GGD

8.8487

0.6605

0.5387

225.932

0.112

0.908

Data 4

GLD

1.8488

0.0245

0.0114

1249.85

0.957

0

GGD

2.5118

0.9386

0.0159

1250.768

0.954

0

Data 5

GLD

1.0932

5.0688

0.0209

788.575

0.439

0

GGD

26.3684

0.2718

7.9396

781.728

0.476

0

Data 6

GLD

17.5655

5.1916

0.6181

147.087

0.067

0.999

GGD

14.9397

1.1094

0.3437

147.092

0.068

0.999

Data 7

GLD

0.0557

5.064

0.0047

744.975

0.169

0.072

GGD

5.5727

0.3912

0.7576

741.716

0.142

0.193

Data 8

GLD

0.0524

5.075

0.0047

564.096

0.15

0.248

GGD

27.7234

0.1822

11.2554

555.636

0.079

0.921

Data 9

GLD

1.1851

0.0006

0.1287

822.169

0.877

0

GGD

3.8869

0.5139

1.3883

816.852

0.873

0

Data 10

GLD

0.8114

0.0007

0.0144

304.348

0.948

0

GGD

6.4942

0.308

2.1333

302.68

0.933

0

Data 11

GLD

1.0628

0.0006

0.5647

110.826

0.936

0

GGD

5.9538

0.3802

5.2747

109.721

0.927

0

Data 12

GLD

2.0093

0.0007

0.2038

634.6

0.043

0.994

GGD

3.8037

0.7017

0.8028

634.035

0.036

0.999

Data 13

GLD

0.9427

0.0003

0.0081

173.873

0.726

0

GGD

26.6637

0.1736

12.7036

170.488

0.726

0

Data 14

GLD

9.6686

0.0029

5.0891

35.637

0.609

0

GGD

51.4619

0.435

39.4639

34.376

0.6

0

Data 15

GLD

17.9881

14.6111

0.615

208.233

0.135

0.58

GGD

19.672

0.9814

0.68

208.225

0.136

0.562

Data 16

GLD

22.7198

4.771

9.3907

101.959

0.056

0.979

GGD

3.5861

2.6483

0.3044

100.581

0.044

0.999

Data 17

GLD

1.2025

0.0832

0.0641

128.161

0.095

0.997

GGD

0.8597

1.4152

0.0068

127.931

0.095

0.997

Data 18

GLD

0.8186

3.974

0.0101

304.883

0.132

0.769

GGD

1.9916

0.9426

0.0152

304.928

0.139

0.719

Table 1: ML Estimates, -2ln L, K-S Statistics and p-values of the fitted distributions of data sets 1 to 18.

The formula for K-S Statistics is defined as follow:

K-S= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4sai aab2cacaWGtbGaeyypa0ZaaCbeaeaacaqGtbGaaeyDaiaabchaaeaa jugWaiaadIhaaKqbagqaamaaemaabaGaamOramaaBaaabaqcLbmaca WGUbaajuaGbeaadaqadaqaaiaadIhaaiaawIcacaGLPaaacqGHsisl caWGgbWaaSbaaeaajugWaiaaicdaaKqbagqaamaabmaabaqcLbmaca WG4baajuaGcaGLOaGaayzkaaaacaGLhWUaayjcSdaaaa@515E@ , where F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOraS WaaSbaaKqbagaajugWaiaad6gaaKqbagqaamaabmaabaGaamiEaaGa ayjkaiaawMcaaaaa@3D3E@ is the empirical distribution function. The best distribution corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ and K-S statistics and higher p-values. It is clear from the goodness of fit of GLD and GGD that in most of the data sets except Data sets (1-18) GGD gives much closer fit than GLD for modeling lifetime data.

0.55

0.93

1.25

1.36

1.49

1.52

1.58

1.61

1.64

1.68

1.73

1.81

2.00

0.74

1.04

1.27

1.39

1.49

1.53

1.59

1.61

1.66

1.68

1.76

1.82

2.01

0.77

1.11

1.28

1.42

1.5

1.54

1.6

1.62

1.66

1.69

1.76

1.84

2.24

0.81

1.13

1.29

1.48

1.51

1.55

1.61

1.62

1.66

1.70

1.77

1.84

0.84

1.24

1.3

1.48

1.51

1.55

1.61

1.63

1.67

1.70

1.78

1.89

Data Set 1: The data set represents the strength of 1.5cm glass fibers measured at the National Physical Laboratory, England. Unfortunately, the units of measurements are not given in the paper, and they are taken from Smith & Naylor [8].

5

25

31

32

34

35

38

39

39

40

42

43

43

43

44

44

47

47

48

49

49

49

51

54

55

55

55

56

56

56

58

59

59

59

59

59

63

63

64

64

65

65

65

66

66

66

66

66

67

67

67

68

69

69

69

69

71

71

72

73

73

73

74

74

76

76

77

77

77

77

77

77

79

79

80

81

83

84

84

86

86

87

90

91

92

92

92

92

93

94

97

98

98

99

101

103

105

109

136

147

Data Set 2: The data is given by Birnbaum & Saunders [9] on the fatigue life of 6061 –T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data (x10-3 ) are presented below (after subtracting 65).

17.88

28.92

33.00

41.52

42.12

45.60

48.80

51.84

51.96

54.12

55.56

67.80

68.44

68.64

68.88

84.12

93.12

98.64

105.12

105.84

127.92

127.92

128.04

173.40

Data Set 3: The data set is from Lawless [10]. The data given arose in tests on endurance of deep groove ball bearings. The data are the number of million revolutions before failure for each of the 23 ball bearings in the life tests and they are:

86

146

251

653

98

249

400

292

131

169

175

176

76

264

15

364

195

262

88

264

157

220

42

321

180

198

38

20

61

121

282

224

149

180

325

250

196

90

229

166

38

337

65

151

341

40

40

135

597

246

211

180

93

315

353

571

124

279

81

186

497

182

423

185

229

400

338

290

398

71

246

185

188

568

55

55

61

244

20

284

393

396

203

829

239

236

286

194

277

143

198

264

105

203

124

137

135

350

193

188

Data Set 4: The data is from Picciotto [11] and arose in test on the cycle at which the Yarn failed. The data are the number of cycles until failure of the yarn and they are:

10

33

44

56

59

72

74

77

92

93

96

100

100

102

105

107

107

108

108

108

109

112

113

115

116

120

121

122

122

124

130

134

136

139

144

146

153

159

160

163

163

168

171

172

176

183

195

196

197

202

213

215

216

222

230

231

240

245

251

253

254

254

278

293

327

342

347

361

402

432

458

555

Data Set 5: This data represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal [12].

19(16)

20(15)

21(14)

22(9)

23(12)

24(10)

25(6)

26(9)

27(8)

28(5)

29(6)

30(4)

31(3)

32(4)

34

35(4)

36(2)

37(2)

39

42

44

Data Set 6: This data is related with behavioral sciences, collected by Balakrishnan N et al. [13]: The scale “General Rating of Affective Symptoms for Preschoolers (GRASP)” measures behavioral and emotional problems of children, which can be classified with depressive condition or not according to this scale. A study conducted by the authors in a city located at the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis, which are:

6.53

7

10.42

14.48

16.10

22.70

34

41.55

42

45.28

49.40

53.62

63

64

83

84

91

108

112

129

133

133

139

140

140

146

149

154

157

160

160

165

146

149

154

157

160

160

165

173

176

218

225

241

248

273

277

297

405

417

420

440

523

583

594

1101

1146

1417

Data Set 7: The data set reported by Efron [14] represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using radiotherapy (RT).

12.20

23.56

23.74

25.87

31.98

37

41.35

47.38

55.46

58.36

63.47

68.46

78.26

74.47

81.43

84

92

94

110

112

119

127

130

133

140

146

155

159

173

179

194

195

209

249

281

319

339

432

469

519

633

725

817

1776

Data Set 8: The data set reported by Efron [14] represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy (RT+CT).

0.08

2.09

3.48

4.87

6.94

8.66

13.11

23.63

0.20

2.23

3.52

4.98

6.97

9.02

13.29

0.40

2.26

3.57

5.06

7.09

9.22

13.80

25.74

0.50

2.46

3.64

5.09

7.26

9.47

14.24

25.82

0.51

2.54

3.70

5.17

7.28

9.74

14.76

6.31

0.81

2.62

3.82

5.32

7.32

10.06

14.77

32.15

2.64

3.88

5.32

7.39

10.34

14.83

34.26

0.90

2.69

4.18

5.34

7.59

10.66

15.96

36.66

1.05

2.69

4.23

5.41

7.62

10.75

16.62

43.01

1.19

2.75

4.26

5.41

7.63

17.12

46.12

1.26

2.83

4.33

5.49

7.66

11.25

17.14

79.05

1.35

2.87

5.62

7.87

11.64

17.36

1.40

3.02

4.34

5.71

7.93

11.79

18.10

1.46

4.40

5.85

8.26

11.98

19.13

1.76

3.25

4.50

6.25

8.37

12.02

2.02

3.31

4.51

6.54

8.53

12.03

20.28

2.02

3.36

6.76

12.07

21.73

2.07

3.36

6.93

8.65

12.63

22.69

Data Set 9: This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee & Wang [15].

23

261

87

7

120

14

62

47

225

71

246

21

42

20

5

12

120

11

3

14

71

11

14

11

16

90 1

16

52

95

Data Set 10: This data set is given by Linhart & Zucchini [16] which represents the failure times of the air conditioning system of an airplane

5.1

1.2

1.3

0.6

0.5

2.4

0.5

1.1

8

0.8

0.4

0.6

0.9

0.4

2

0.5

5.3

3.2

2.7

2.9

2.5

2.3

1

0.2

0.1

0.1

1.8

0.9

2

4

6.8

1.2

0.4

0.2

 

 

Data Set 11: This data set used by Bhaumik et al. [17], is vinyl chloride data obtained from clean up gradient monitoring wells in mg/l

0.8

0.8

1.3

1.5

1.8

1.9

1.9

2.1

2.6

2.7

2.9

3.1

3.2

3.3

3.5

3.6

4

4.1

4.2

4.2

4.3

4.3

4.4

4.4

4.6

4.7

4.7

4.8

4.9

4.9

5

5.3

5.5

5.7

5.7

6.1

6.2

6.2

6.2

6.3

6.7

6.9

7.1

7.1

7.1

7.1

7.4

7.6

7.7

8

8.2

8.6

8.6

8.6

8.8

8.8

8.9

8.9

9.5

9.6

9.7

9.8

10.7

10.9

11

11

11.1

11.2

11.2

11.5

11.9

12.4

12.5

12.9

13

13.1

13.3

13.6

13.7

13.9

14.1

15.4

15.4

17.3

17.3

18.1

18.2

18.4

18.9

19

19.9

20.6

21.3

21.4

21.9

23

27

31.6

33.1

38.5

Data Set 12: This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al. [4] for fitting the Lindley [3] distribution.

74

57

48

29

502

12

70

21

29

386

59

27

153

26

326

Data Set 13: This data is for the times between successive failures of air conditioning equipment in a Boeing 720 airplane, Proschan [18].

1.1

1.4

1.3

1.7

1.9

1.8

1.6

2.2

1.7

2.7

4.1

1.8

1.5

1.2

1.4

3

1.7

2.3

1.6

2

Data Set 14: This data set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross & Clark [19].

18.83

20.8

21.657

23.03

23.23

24.05

24.321

25.5

25.52

25.8

26.69

26.77

26.78

27.05

27.67

29.9

31.11

33.2

33.73

33.76

33.89

34.76

35.75

35.91

36.98

37.08

37.09

39.58

44.045

45.29

45.381

Data Set 15: This data set is the strength data of glass of the aircraft window reported by Fuller et al. [20].

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

Data Set 16: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm, Bader & Priest [21].

1.4

5.1

6.3

10.8

12.1

18.5

19.7

22.2

23.0

30.6

37.3

46.3

53.9

59.8

66.2

Data Set 17: The following data set represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test, Lawless [10].

15

20

38

42

61

76

86

98

121

146

149

157

175

176

180

180

198

220

224

251

264

282

321

325

653

Data Set 18: The following data set represents the number of cycles to failure for 25 100-cm specimens of yarn, tested at a particular strain level, Lawless [10].

Conclusion

The modeling and analysis of lifetime data using lifetime distributions are crucial in almost all applied sciences including behavioral sciences, medicine, insurance, engineering, and finance, amongst others. In this paper an attempt has been made to have a comparative study on modeling of lifetime data on eighteen data sets using three parameters generalized Lindley distribution (GLD) introduced by Zakerzadeh & Dolati [1] and generalized gamma distribution (GGD) introduced by Stacy [2]. Maximum likelihood estimates have been used for fitting both GLD and GGD. The goodness of fit for both GLD and GGD shows that GGD gives much closer fit than GLD in majority of data sets and hence GGD can be considered as an important tool for modeling lifetime data over GLD.

References

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  2. Stacy EW (1962) A generalization of the gamma distribution. Annals of Mathematical Statistical 33: 1187-1192.
  3. Lindley DV (1958) Fiducial distributions and Bayes’ Theorem. Journal of the Royal Statistical Society Series B 20(1): 102-107.
  4. Ghitany ME, Atieh B, Nadarajah S (2008) Lindley distribution and its Application. Mathematics Computing and Simulation 78(4): 493-506.
  5. Shanker R, Hagos F, Sujatha S (2015) On modeling of Lifetimes data using exponential and Lindley distributions. Biometrics & Biostatistics International Journal 2(5): 1-9.
  6. Shanker R (2016) On generalized Lindley distribution and its applications, To appear in, “Insights in Biomedicine”
  7. Stacy EW, Mihram GA (1965) Parametric estimation for a generalized gamma distribution. Technometrics 7: 349-358.
  8. Smith RL, Naylor JC (1987) A comparison of Maximum likelihood and Bayesian estimators for the three parameter Weibull distribution Applied Statistics 36(3): 358-369.
  9. Birnbaum ZW, Saunders SC (1969) Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability 6(2): 328-347.
  10. Lawless JF (2003) Statistical models and methods for lifetime data. John Wiley and Sons, New York, USA.
  11. Picciotto R (1970) Tensile fatigue characteristics of a sized polyester/viscose yarn and their effect on weaving performance. Master thesis, North Carolina State, USA.
  12. Bjerkedal T (1960) Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. American Journal of Epidemiology 72(1): 130-148.
  13. Balakrishnan N, Victor L, Antonio S (2010) A mixture model based on Birnhaum-Saunders Distributions, A study conducted by Authors regarding the Scores of the GRASP (General Rating of Affective Symptoms for Preschoolers), in a city located at South Part of the Chile.
  14. Efron B (1988) Logistic regression, survival analysis and the Kaplan-Meier curve. Journal of the American Statistical Association 83(408): 414-425.
  15. Lee ET, Wang JW (2003) Statistical methods for survival data analysis. 3rd edition, John Wiley and Sons, New York, USA.
  16. Linhart H, Zucchini W (1986) Model Selection, John Wiley, New York, USA.
  17. Bhaumik DK, Kapur K, Gibbons RD (2009) Testing Parameters of a Gamma Distribution for Small Samples. Technometrics 51(3): 326-334.
  18. Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 5(3): 375-383.
  19. Gross AJ, Clark VA (1975) Survival Distributions: Reliability Applications in the Biometrical Sciences, John Wiley, New York, USA.
  20. Fuller EJ, Frieman S, Quinn J, Quinn G, Carter W (1994) Fracture mechanics approach to the design of glass aircraft windows: A case study. SPIE Proc 2286: 419-430.
  21. Bader MG, Priest AM (1982) Statistical aspects of fiber and bundle strength in hybrid composites In: hayashi T, Kawata, K Umekawa S (Eds.), Progressin Science in                          Engineering Composites, Tokyo, Japan: 1129-1136.
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