ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 5 - 2016
On Modeling of Lifetime Data Using Two-Parameter Gamma and Weibull Distributions
Rama Shanker1*, Kamlesh Kumar Shukla1, Ravi Shanker2 and Tekie Asehun Leonida3
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Mathematics, GLA College, NP University, India
3Department of Applied Mathematics, University of Twente, The Netherlands
Received: September 19, 2016| Published: October 07, 2016
*Corresponding author: Rama Shanker and Kamlesh Kumar Shukla, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email: ;
Citation: Shanker R, Shukla KK, Shanker R, Leonida TS (2016) On Modeling of Lifetime Data Using Two-Parameter Gamma and Weibull Distributions. Biom Biostat Int J 4(5): 00107. DOI: 10.15406/bbij.2016.04.00107

Abstract

The analysis and modeling of lifetime data are crucial in almost all applied sciences including medicine, insurance, engineering, behavioral sciences and finance, amongst others. The main objective of this paper is to have a comparative study of two-parameter gamma and Weibull distributions for modeling lifetime data from various fields of knowledge. Since exponential distribution is a particular case of both gamma and Weibull distributions and the exponential distribution is a classical distribution for modeling lifetime data, the goodness of fit of both gamma and Weibull distributions are compared with exponential distribution.

Keywords: Gamma distribution; Weibull distribution; Exponential distribution; Lifetime data; Estimation of parameter; Goodness of fit

Introduction

The lifetime or survival time or failure time in reliability analysis is the time to the occurrence of event of interest. 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 modeling and statistical analysis 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.

The statistics literature is flooded with lifetime distributions including exponential distribution, gamma distribution, Lindley distribution, Weibull distribution and their generalizations, some amongst others.

Gamma Distribution

The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of two-parameter gamma distribution (GD) having parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ are given by

f 1 ( x;θ,α )= θ α Γ( α ) x α1 e θx ;x>0,θ>0,α>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXjaacYcacqaHXoqyaiaawIcacaGLPaaacqGH9aqpdaWcaaqaai abeI7aXnaaCaaabeqcfasaaiabeg7aHbaaaKqbagaacqqHtoWrdaqa daqaaiabeg7aHbGaayjkaiaawMcaaaaacaWG4bWaaWbaaeqajuaiba GaeqySdeMaeyOeI0IaaGymaaaajuaGcaaMc8UaamyzamaaCaaabeqc fasaaiabgkHiTiabeI7aXjaaykW7caWG4baaaKqbakaacUdacaaMc8 UaaGPaVlaadIhacqGH+aGpcaaIWaGaaiilaiaaykW7cqaH4oqCcqGH +aGpcaaIWaGaaiilaiaaykW7cqaHXoqycqGH+aGpcaaIWaaaaa@69AD@ (2.1)

F 2 ( x;θ,α )=1 Γ( α,θx ) Γ( α ) ;x>0,θ>0,α>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXjaacYcacqaHXoqyaiaawIcacaGLPaaacqGH9aqpcaaIXaGaey OeI0YaaSaaaeaacqqHtoWrdaqadaqaaiabeg7aHjaacYcacqaH4oqC caWG4baacaGLOaGaayzkaaaabaGaeu4KdC0aaeWaaeaacqaHXoqyai aawIcacaGLPaaaaaGaaGPaVlaaykW7caGG7aGaamiEaiabg6da+iaa icdacaGGSaGaeqiUdeNaeyOpa4JaaGimaiaacYcacaaMc8UaeqySde MaeyOpa4JaaGimaaaa@5FCB@ (2.2)

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 as

Γ( α,z )= z e y y α1 dy;α>0,z0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeu4KdC 0aaeWaaeaacqaHXoqycaGGSaGaamOEaaGaayjkaiaawMcaaiabg2da 9maapehabaGaamyzamaaCaaabeqcfasaaiabgkHiTiaadMhaaaqcfa OaaGPaVlaadMhadaahaaqabKqbGeaacqaHXoqycqGHsislcaaIXaaa aKqbakaaykW7caWGKbGaamyEaiaaykW7caGG7aGaeqySdeMaeyOpa4 JaaGimaiaacYcacaaMc8UaaGPaVlaadQhacqGHLjYScaaIWaaajuai baGaamOEaaqaaiabg6HiLcqcfaOaey4kIipaaaa@5E09@ (2.3)

It can be easily shown that the gamma distribution reduces to classical exponential distribution for α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde Maeyypa0JaaGymaaaa@39E4@ having p.d.f. and c.d.f.

f 2 ( x;θ )=θ e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iabeI7aXjaaykW7caWGLbWaaW baaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlaadIhaaaqcfaOaai4o aiaaykW7caWG4bGaeyOpa4JaaGimaiaacYcacqaH4oqCcqGH+aGpca aIWaaaaa@526E@ (2.4)

F 2 ( x;θ )=1 e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXbGaayjkaiaawMcaaiabg2da9iaaigdacqGHsislcaWGLbWaaW baaeqajuaibaGaeyOeI0IaeqiUdeNaamiEaaaajuaGcaaMc8Uaai4o aiaadIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6da+iaaicdaaa a@4F2A@ (2.5)

It should be noted that the gamma distribution is the weighted exponential distribution. Stacy [1] obtained the generalization of the gamma distribution. Stacy & Mihram [2] have detailed discussion about parametric estimation of generalized gamma distribution.

Weibull Distribution

The p.d.f. and the c.d.f. of two-parameter Weibull distribution having parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde gaaa@3823@ are given by

f 3 ( x;θ,α )=θα x α1 e θ x α ;x>0,θ>0,α>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXjaacYcacqaHXoqyaiaawIcacaGLPaaacqGH9aqpcqaH4oqCca aMc8UaeqySdeMaamiEamaaCaaabeqcfasaaiabeg7aHjabgkHiTiaa igdaaaqcfaOaaGPaVlaadwgadaahaaqabKqbGeaacqGHsislcqaH4o qCcaaMc8UaamiEaKqbaoaaCaaajuaibeqaaiabeg7aHbaaaaqcfaOa aGPaVlaaykW7caGG7aGaamiEaiabg6da+iaaicdacaGGSaGaeqiUde NaeyOpa4JaaGimaiaacYcacqaHXoqycqGH+aGpcaaIWaaaaa@6523@ (3.1)

F 3 ( x;θ,α )=1 e θ x α ;x>0,θ>0,α>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaaG4maaqcfayabaWaaeWaaeaacaWG4bGaai4oaiab eI7aXjaacYcacqaHXoqyaiaawIcacaGLPaaacqGH9aqpcaaIXaGaey OeI0IaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaadIhajuaG daahaaqcfasabeaacqaHXoqyaaaaaKqbakaaykW7caaMc8Uaai4oai aadIhacqGH+aGpcaaIWaGaaiilaiabeI7aXjabg6da+iaaicdacaGG SaGaeqySdeMaeyOpa4JaaGimaaaa@5993@ (3.2)

It can be easily shown that the Weibull distribution reduces to classical exponential distribution at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqySde Maeyypa0JaaGymaaaa@39E4@ . It should be noted that Weibull distribution is nothing but the power exponential distribution.

Taking x= y 1 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiEai abg2da9iaadMhadaahaaqabKqbGeaajuaGdaWcaaqcfasaaiaaigda aeaacqaHXoqyaaaaaaaa@3CFB@ and thus y= x α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyEai abg2da9iaadIhadaahaaqabKqbGeaacqaHXoqyaaaaaa@3B74@ in (2.4), we have

g( y;θ,α )=f( y α ) f ( y )=θ e θ y α α y α1 =θα y α1 e θ y α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4zam aabmaabaGaamyEaiaacUdacqaH4oqCcaGGSaGaeqySdegacaGLOaGa ayzkaaGaeyypa0JaamOzamaabmaabaGaamyEamaaCaaabeqcfasaai abeg7aHbaaaKqbakaawIcacaGLPaaaceWGMbGbauaadaqadaqaaiaa dMhaaiaawIcacaGLPaaacqGH9aqpcqaH4oqCcaWGLbWaaWbaaeqaju aibaGaeyOeI0IaeqiUdeNaaGPaVlaadMhajuaGdaahaaqcfasabeaa cqaHXoqyaaaaaKqbakabeg7aHjaadMhadaahaaqabKqbGeaacqaHXo qycqGHsislcaaIXaaaaKqbakabg2da9iabeI7aXjaaykW7cqaHXoqy caWG5bWaaWbaaeqajuaibaGaeqySdeMaeyOeI0IaaGymaaaajuaGca WGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlaadMhajuaG daahaaqcfasabeaacqaHXoqyaaaaaaaa@7009@

Which is the p.d.f. of Weibull distribution defined in (3.1)

Maximum Likelihood Estimation

Maximum likelihood estimates of the parameters of gamma distribution (GD): Assuming ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaaGPaVlaa dIhadaWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaaMc8UaamiEam aaBaaajuaibaGaaG4maaqcfayabaGaaiilaiaaykW7caaMc8UaaiOl aiaac6cacaGGUaGaaGPaVlaaykW7caGGSaGaamiEamaaBaaajuaiba GaamOBaaqcfayabaaacaGLOaGaayzkaaaaaa@50B4@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@ from Gamma distribution (2.1), the likelihood function is given by

L= ( θ α Γ( α ) ) n ( i=1 n x i ) α1 e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacqaH XoqyaaaajuaGbaGaeu4KdC0aaeWaaeaacqaHXoqyaiaawIcacaGLPa aaaaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaamOBaaaajuaGdaqa daqaamaarahabaGaamiEamaaBaaajuaibaGaamyAaaqcfayabaaaju aibaGaamyAaiabg2da9iaaigdaaeaacaWGUbaajuaGcqGHpis1aaGa ayjkaiaawMcaamaaCaaabeqcfasaaiabeg7aHjabgkHiTiaaigdaaa qcfaOaamyzamaaCaaabeqcfasaaiabgkHiTiaad6gacaaMc8UaeqiU deNaaGPaVlqadIhagaqeaaaaaaa@5C70@ , 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, ln L of Gamma distribution is thus given by

lnL=n[ αlnθln( Γ( α ) ) ]+( α1 ) i=1 n ln x i nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBamaadmaabaGaeqySdeMaciiBaiaa c6gacqaH4oqCcqGHsislciGGSbGaaiOBamaabmaabaGaeu4KdC0aae WaaeaacqaHXoqyaiaawIcacaGLPaaaaiaawIcacaGLPaaaaiaawUfa caGLDbaacqGHRaWkdaqadaqaaiabeg7aHjabgkHiTiaaigdaaiaawI cacaGLPaaadaaeWbqaaiGacYgacaGGUbGaamiEamaaBaaajuaibaGa amyAaaqcfayabaaajuaibaGaamyAaiabg2da9iaaigdaaeaacaWGUb aajuaGcqGHris5aiabgkHiTiaad6gacaaMc8UaeqiUdeNaaGPaVlqa dIhagaqeaaaa@63EE@

The maximum likelihood estimate (MLE) θ ^ and α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7caaMc8Ua fqySdeMbaKaaaaa@42E1@ of parameters θandα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaaGPaVlaaykW7caqGHbGaaeOBaiaabsgacaaMc8UaaGPaVlabeg7a Hbaa@42C1@ of gamma distribution can be obtained by solving the natural log likelihood equation using R software (Package Stat 4).

Maximum likelihood estimates of the parameters of weibull distribution

Assuming ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG4bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaaGPaVlaa dIhadaWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaaMc8UaamiEam aaBaaajuaibaGaaG4maaqcfayabaGaaiilaiaaykW7caaMc8UaaiOl aiaac6cacaGGUaGaaGPaVlaaykW7caGGSaGaamiEamaaBaaajuaiba GaamOBaaqcfayabaaacaGLOaGaayzkaaaaaa@50B4@ be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@ from GD (3.1), the natural log likelihood function, ln of Weibull distribution is given by

lnL= i=1 n lnf( x i ;θ,α ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0ZaaabCaeaaciGGSbGaaiOBaiaadAgadaqa daqaaiaadIhadaWgaaqcfasaaiaadMgaaKqbagqaaiaacUdacqaH4o qCcaGGSaGaeqySdegacaGLOaGaayzkaaaajuaibaGaamyAaiabg2da 9iaaigdaaeaacaWGUbaajuaGcqGHris5aaaa@4CAB@ =n( lnθ+lnα )+( α1 ) i=1 n ln x i θ i=1 n x i α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyypa0 JaamOBamaabmaabaGaciiBaiaac6gacqaH4oqCcqGHRaWkciGGSbGa aiOBaiabeg7aHbGaayjkaiaawMcaaiabgUcaRmaabmaabaGaeqySde MaeyOeI0IaaGymaaGaayjkaiaawMcaamaaqahabaGaciiBaiaac6ga caWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaaaKqbGeaacaWGPbGaey ypa0JaaGymaaqaaiaad6gaaKqbakabggHiLdGaeyOeI0IaeqiUde3a aabCaeaacaWG4bWaaSbaaKqbGeaacaWGPbaajuaGbeaadaahaaqabK qbGeaacqaHXoqyaaaabaGaamyAaiabg2da9iaaigdaaeaacaWGUbaa juaGcqGHris5aaaa@609F@

The maximum likelihood estimate (MLE) θ ^ and α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacaaMc8UaaGPaVlaabggacaqGUbGaaeizaiaaykW7caaMc8Ua fqySdeMbaKaaaaa@42E1@ of parameters θandα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaaGPaVlaaykW7caqGHbGaaeOBaiaabsgacaaMc8UaaGPaVlabeg7a Hbaa@42C1@ of Weibull distribution 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 gamma and Weibull distributions discussed for several lifetime data and fit is compared with exponential distribution. In order to compare gamma, Weibull, and exponential distributions, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ and K-S Statistics ( Kolmogorov-Smirnov Statistics) for fifteen data sets have been computed and presented in Table 1. 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 aab2cacaWGtbGaeyypa0ZaaCbeaeaacaqGtbGaaeyDaiaabchaaKqb GeaacaWG4baajuaGbeaadaabdaqaaiaadAeadaWgaaqcfasaaiaad6 gaaKqbagqaamaabmaabaGaamiEaaGaayjkaiaawMcaaiabgkHiTiaa dAeadaWgaaqcfasaaiaaicdaaKqbagqaamaabmaabaGaamiEaaGaay jkaiaawMcaaaGaay5bSlaawIa7aaaa@4CA2@ , where F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamOBaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaaaaa@3BA5@ 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.

Distribution

ML Estimates

-2In L

K-S Statistics

P-value

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

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

Data 1

Gamma

11.5711

17.4355

47.903

0.809

0.000

Weibull

0.0598

5.7796

30.413

0.803

0.000

Exponential

0.6636

177.660

0.564

0.000

Data 2

Gamma

0.0558

4.0280

226.045

0.123

0.838

Weibull

0.0021

1.4377

232.269

0.229

0.152

Exponential

0.0138

242.870

0.307

0.019

Data 3

Gamma

0.0209

2.0833

788.495

0.996

0.000

Weibull

0.0029

1.2849

795.750

0.177

0.021

Exponential

0.0057

889.220

0.297

0.000

Data 4

Gamma

0.0046

1.0320

744.834

0.166

0.079

Weibull

0.0059

0.9521

744.845

0.151

0.139

Exponential

0.0045

744.881

0.16

0.101

Data 5

Gamma

0.0047

1.0476

564.029

0.148

0.259

Weibull

0.0064

0.9404

563.68

0.129

0.419

Exponential

0.0045

564.03

0.139

0.33

Data 6

Gamma

0.1287

1.1851

822.169

0.878

0.000

Weibull

0.0946

1.0514

823.785

0.873

0.000

Exponential

0.1085

824.371

0.868

0.000

Data 7

Gamma

0.0136

0.8127

304.335

0.947

0.000

Weibull

0.0329

0.853

303.874

0.944

0.000

Exponential

0.0167

305.25

0.954

0.000

Data 8

Gamma

0.5654

1.0627

110.826

0.937

0.000

Weibull

0.5263

1.0102

110.899

0.934

0.000

Exponential

0.532

110.901

0.934

0.000

Data 9

Gamma

0.2034

2.0095

634.6

0.043

0.993

Weibull

0.0306

1.4573

637.461

0.057

0.9

Exponential

0.1012

658.041

0.173

0.005

Data 10

Gamma

0.0076

0.9157

173.852

0.719

0.000

Weibull

0.0032

1.1731

175.978

0.797

0.000

Exponential

0.0083

173.94

0.74

0.000

Data 11

Gamma

5.0874

9.6662

35.637

0.609

0.000

Weibull

0.1215

2.7869

41.173

0.587

0.000

Exponential

0.5263

65.67

0.471

0.000

Data 12

Gamma

0.6146

18.9374

208.231

0.135

0.577

Weibull

0.0021

1.8108

241.63

0.368

0.000

Exponential

0.0325

274.531

0.458

0.000

Data 13

Gamma

9.2878

22.8042

101.971

0.057

0.979

Weibull

0.0065

5.1692

103.482

0.066

0.917

Exponential

0.4079

261.701

0.448

0.000

Data 14

Gamma

0.0523

1.4412

128.372

0.102

0.992

Weibull

0.0123

1.2978

128.041

0.099

0.995

Exponential

0.0363

129.47

0.156

0.807

Data 15

Gamma

0.0101

1.8082

304.876

0.136

0.748

Weibull

0.0027

1.1423

306.687

0.191

0.32

Exponential

0.0056

309.181

0.202

0.257

Table 1: ML Estimates, -2ln L, K-S Statistics and p-values of the fitted distributions of data sets 1 to 15

From table 1 it is clear that gamma distribution gives better fit in data sets 2,3,4,6,8,10,11,12,13, and 15 while Weibull distribution gives better fit in data sets 1,5,7,9, and 14 Data sets (1-15).

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

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

1.5

1.55

1.61

1.62

1.66

1.7

1.77

1.84

0.84

1.24

1.3

1.48

1.51

1.55

1.61

1.63

1.67

1.7

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 [3].

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

128.04

173.40

Data Set 2: The data set is from Lawless [4]. 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:

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

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 4: The data set reported by Efron [6] 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 5: The data set reported by Efron [6] 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

0.2

2.23

3.52

4.98

6.97

9.02

13.29

0.40

2.26

3.57

5.06

7.09

9.22

13.8

25.74

0.50

2.46

3.64

5.09

7.26

9.47

14.24

25.82

0.51

2.54

3.7

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 6: This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee & Wang [7].

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 7: This data set is given by Linhart & Zucchini [8], 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 8: This data set used by Bhaumik et al. [9], is vinyl chloride data obtained from clean upgradient 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

4.0

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

5.0

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

8.0

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

11.0

11.0

11.1

11.2

11.2

11.5

11.9

12.4

12.5

12.9

13.0

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

19.0

19.9

20.6

21.3

21.4

21.9

23.0

27.0

31.6

33.1

38.5

Data set 9: This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al. [10] for fitting the Lindley [11] distribution.

74

57

48

29

502

12

70

21

29

386

59

27

153

26

326

Data Set 10: This data is for the times between successive failures of air conditioning equipment in a Boeing 720 airplane, Proschan [12].

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 11: 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 [13].

18.83

20.8

21.66

23.03

23.23

24.05

24.321

25.5

25.5

25.8

26.69

26.77

26.78

27.05

27.67

29.9

31.11

33.2

33.73

33.8

33.9

34.76

35.75

35.91

36.98

37.08

37.09

39.58

44.05

45.29

45.381

Data Set 12: This data set is the strength data of glass of the aircraft window reported by Fuller et al. [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

3.858

 

 

 

Data Set 13: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm, Bader & Priest [15].

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 14: The following data set represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test, Lawless [4].

15

20

38

42

61

76

86

98

121

146

149

157

175

176

180

180

198

220

224

251

264

282

321

325

635
                     

Data Set 15: 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 [4].

Concluding Remarks

In this paper an attempt has been made to have the comparative and detailed study of two-parameter gamma and Weibull distributions for modeling lifetime data from various fields of knowledge. Since exponential distribution is a particular case of both gamma and Weibull distributions and the exponential distribution is a classical distribution for modeling lifetime data, the goodness of fit of both gamma and Weibull distributions are compared with exponential distribution. From the fitting of exponential, Weibull and gamma distribution it is obvious that in majority of data sets gamma distribution gives better fit than both Weibull and exponential distribution.

References

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