ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Review Article
Volume 2 Issue 5 - 2014
On Modeling of Lifetimes Data Using Exponential and Lindley Distributions
Rama Shanker1*, Hagos Fesshaye2 and Sujatha Selvaraj3
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
3Department of Banking and Finance, Jimma University, Ethiopia
Received: June 15, 2015 | Published: June 26, 2015
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H, Selvaraj S (2015) On Modeling of Lifetimes Data Using Exponential and Lindley Distributions. Biom Biostat Int J 2(5): 00042. DOI: 10.15406/bbij.2015.02.00042

Abstract

In this paper, firstly the nature of exponential and Lindley distributions have been studied using different graphs of their probability density functions and cumulative distribution functions. The expressions for the index of dispersion for both exponential and Lindley distributions have been obtained and the conditions under which the exponential and Lindley distributions are over-dispersed, equi-dispersed, and under-dispersed has been given. Several real lifetimes data-sets has been fitted using exponential and Lindley distributions for comparative study and it has been shown that in some cases exponential distribution provides better fit than the Lindley distribution whereas in other cases Lindley distribution provides better fit than the exponential distribution.

Keywords: Exponential distribution; Lindley distribution; Index of dispersion; Estimation of parameter; Goodness of fit

Introduction

The time to the occurrence of some event is of interest for some populations of individuals in every field of knowledge. The event may be death of a person, failure of a piece of equipment, development of (or remission) of symptoms, health code violation (or compliance). The times to the occurrences of events are known as “lifetimes” or “survival times” or “failure times” according to the event of interest in the fields of study. The statistical analysis of lifetime data has been a topic of considerable interest to statisticians and research workers in areas such as engineering, medical and biological sciences. Applications of lifetime distributions range from investigations into the endurance of manufactured items in engineering to research involving human diseases in biomedical sciences.

There are a number of continuous distributions for modeling lifetime data such as exponential, Lindley, gamma, lognormal and Weibull. The exponential, Lindley and the Weibull distributions are more popular in practice than the gamma and the lognormal distributions because the survival functions of the gamma and the lognormal distributions cannot be expressed in closed forms and both require numerical integration. Both exponential and Lindley distributions are of one parameter and the Lindley distribution has advantage over the exponential distribution that the exponential distribution has constant hazard rate and mean residual life function whereas the Lindley distribution has increasing hazard rate and decreasing mean residual life function.

In this paper, firstly the nature of exponential and Lindley distribution has been studied by drawing different graphs for probability densities and cumulative distribution functions for the same values of parameter. Several examples of lifetimes data-sets from different fields of knowledge has been considered and an attempt has been made to study the goodness-of- fit for both exponential and Lindley distributions to see the superiority of one over the other.

Exponential and Lindley Distributions

Exponential Distribution
 The exponential distribution was the first widely used lifetime distribution model in areas ranging from studies on the lifetimes of manufactured items [1-3] to research involving survival or remission times in chronic diseases [4]. The main reason for its wide applicability as lifetime model is partly because of the availability of simple statistical methods for it [2] and partly because it appeared suitable for representing the lifetimes of many things such as various types of manufactured items [1].

Let T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36CF@ be a continuous random variable representing the lifetimes of individuals in some population and following exponential distribution. The probability density function (p.d.f.), cumulative distribution function (c.d.f.), survival function, hazard function, and mean residual life function of T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36CF@ , respectively, are given by
f( t )=θ e θx ;θ>0,t>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcqaH4oqCcaWGLbqcfa4aaWbaaSqabeaajugibiabgkHiTiabeI 7aXjaadIhaaaGaai4oaiabeI7aXjabg6da+iaaicdacaGGSaGaamiD aiabg6da+iaaicdaaaa@4BE4@
F( t )=1 e θt ;θ>0,t>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsislcaWG LbWaaWbaaSqabeaacqGHsislcqaH4oqCcaWG0baaaOGaaGPaVlaayk W7caaMc8Uaai4oaiabeI7aXjabg6da+iaaicdacaGGSaGaaGPaVlaa ykW7caaMc8UaamiDaiabg6da+iaaicdaaaa@5192@
S( t )=1F( t )= e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsislcaWG gbWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyypa0JaamyzamaaCa aaleqabaGaeyOeI0IaeqiUdeNaamiDaaaaaaa@4504@
h( t )= f( t ) 1F( t ) = f( t ) S( t ) =θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaamOzamaa bmaabaGaamiDaaGaayjkaiaawMcaaaqaaiaaigdacqGHsislcaWGgb WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaiabg2da9maalaaabaGa amOzamaabmaabaGaamiDaaGaayjkaiaawMcaaaqaaiaadofadaqada qaaiaadshaaiaawIcacaGLPaaaaaGaeyypa0JaeqiUdehaaa@4D76@
< m( t )= 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqa aiabeI7aXbaaaaa@3CF1@

Lindley distribution
Lindley distribution is a mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH4oqCaiaawIcacaGLPaaaaaa@3935@ and gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3AA1@ distributions with mixing proportion θ θ+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aH4oqCaeaacqaH4oqCcqGHRaWkcaaIXaaaaaaa@3B0F@  and is given by Lindley (1958) in the context of Bayesian Statistics as a counter example of fiducial Statistics. Let T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36CF@ be a continuous random variable representing the lifetimes of individuals in some population and following Lindley distribution. The probability density function (p.d.f.), cumulative distribution function (c.d.f.), survival function, hazard function, and mean residual life function of T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36CF@ , respectively, are given by
f( t )= θ 2 θ+1 ( 1+t ) e θt ;θ>0,t>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIYaaaaaGcbaGaeqiUdeNaey4kaSIaaGymaaaada qadaqaaiaaigdacqGHRaWkcaWG0baacaGLOaGaayzkaaGaamyzamaa CaaaleqabaGaeyOeI0IaeqiUdeNaamiDaaaakiaacUdacaaMc8UaaG PaVlaaykW7cqaH4oqCcqGH+aGpcaaIWaGaaiilaiaaykW7caaMc8Ua amiDaiabg6da+iaaicdaaaa@58AA@
F( t )=1 θ+1+θt θ+1 e θt ;θ>0,t>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWc aaqaaiabeI7aXjabgUcaRiaaigdacqGHRaWkcqaH4oqCcaWG0baaba GaeqiUdeNaey4kaSIaaGymaaaacaWGLbWaaWbaaSqabeaacqGHsisl cqaH4oqCcaWG0baaaOGaaGPaVlaaykW7caaMc8Uaai4oaiabeI7aXj abg6da+iaaicdacaGGSaGaaGPaVlaaykW7caaMc8UaamiDaiabg6da +iaaicdaaaa@5BD9@
S( t )=1F( t )= θ+1+θt θ+1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsislcaWG gbWaaeWaaeaacaWG0baacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacq aH4oqCcqGHRaWkcaaIXaGaey4kaSIaeqiUdeNaamiDaaqaaiabeI7a XjabgUcaRiaaigdaaaGaamyzamaaCaaaleqabaGaeyOeI0IaeqiUde NaamiDaaaaaaa@4F4B@
h( t )= f( t ) 1F( t ) = f( t ) S( t ) = θ 2 ( 1+t ) θ+1+θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaamOzamaa bmaabaGaamiDaaGaayjkaiaawMcaaaqaaiaaigdacqGHsislcaWGgb WaaeWaaeaacaWG0baacaGLOaGaayzkaaaaaiabg2da9maalaaabaGa amOzamaabmaabaGaamiDaaGaayjkaiaawMcaaaqaaiaadofadaqada qaaiaadshaaiaawIcacaGLPaaaaaGaeyypa0ZaaSaaaeaacqaH4oqC daahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaaigdacqGHRaWkcaWG0b aacaGLOaGaayzkaaaabaGaeqiUdeNaey4kaSIaaGymaiabgUcaRiab eI7aXjaadshaaaaaaa@597C@
m( t )= θ+2+θt θ( θ+1+θt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUdeNa ey4kaSIaaGOmaiabgUcaRiabeI7aXjaadshaaeaacqaH4oqCdaqada qaaiabeI7aXjabgUcaRiaaigdacqGHRaWkcqaH4oqCcaWG0baacaGL OaGaayzkaaaaaaaa@4B88@
The Lindley distribution has been extensively studied and generalized by many researchers such as [5-12] are among others. A discrete version of the Lindley distribution has been obtained by [13-14] obtained the Lindley mixture of Poisson distribution.

The graphs of the probability densities functions of exponential and Lindley distributions are presented for different values of parameter and shown in Figure 1. The graphs of the cumulative distribution functions of exponential and Lindley distributions are presented for different values of parameter and are shown in Figure 2.

The expressions for coefficient of variation (C.V.), coefficient of Skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada Gcaaqaaiabek7aInaaBaaaleaacaaIXaaabeaaaeqaaaGccaGLOaGa ayzkaaaaaa@3A21@ , coefficient of Kurtosis, and index of dispersion of exponential and Lindley distributions are summarized in the following Table 1. It can be easily verified that the Lindley distribution is over- dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH8aapcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CEF@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH9aqpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF1@  and under-dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH+aGpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF3@  for θ<(=)> θ =1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ipaWJaaiikaiabg2da9iaacMcacqGH+aGpcqaH4oqCdaahaaWcbeqa aiabgEHiQaaakiabg2da9iaaigdacaGGUaGaaGymaiaaiEdacaaIWa GaaGimaiaaiIdacaaI2aGaaGinaiaaiIdacaaI3aaaaa@4819@  respectively, whereas as exponential distribution is over- dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH8aapcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CEF@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH9aqpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF1@  and under- dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH+aGpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF3@  for θ<(=)> θ =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ipaWJaaiikaiabg2da9iaacMcacqGH+aGpcqaH4oqCdaahaaWcbeqa aiabgEHiQaaakiabg2da9iaaigdaaaa@40B4@  respectively

Applications

The exponential and Lindley distribution has been fitted to a number of real lifetime data - sets to tests their goodness of fit. Goodness of fit tests for fifteen real lifetime data- sets has been presented here.

In order to compare exponential and Lindley distributions, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiGacYgacaGGUbGaamitaaaa@3A54@ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics ( Kolmogorov-Smirnov Statistics) for all fifteen real lifetime data- sets have been computed. The formulae for computing AIC, AICC, BIC, and K-S Statistics are as follows:
AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadM eacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamitaiab gUcaRiaaikdacaWGRbaaaa@4044@ , AICC=AIC+ 2k( k+1 ) ( nk1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadM eacaWGdbGaam4qaiabg2da9iaadgeacaWGjbGaam4qaiabgUcaRmaa laaabaGaaGOmaiaadUgadaqadaqaaiaadUgacqGHRaWkcaaIXaaaca GLOaGaayzkaaaabaWaaeWaaeaacaWGUbGaeyOeI0Iaam4AaiabgkHi TiaaigdaaiaawIcacaGLPaaaaaaaaa@4931@ , BIC=2lnL+klnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaadM eacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamitaiab gUcaRiaadUgaciGGSbGaaiOBaiaadYeaaaa@423E@  and
D= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9maaxababaGaae4uaiaabwhacaqGWbaaleaacaWG4baabeaakmaa emaabaGaamOramaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiEaa GaayjkaiaawMcaaiabgkHiTiaadAeadaWgaaWcbaGaaGimaaqabaGc daqadaqaaiaadIhaaiaawIcacaGLPaaaaiaawEa7caGLiWoaaaa@4890@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ the number of parameters, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ is the sample size and F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGUbaabeaakmaabmaabaGaamiEaaGaayjkaiaawMcaaaaa @3A70@ is the empirical distribution function. The best distribution corresponds to lower 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiGacYgacaGGUbGaamitaaaa@3A54@ , AIC, AICC, BIC, and K-S statistics.

The fittings of exponential and Lindley distributions are based on maximum likelihood estimates (MLE). Let t 1 , t 2 ,...., t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaaabeaakiaacYcacaWG0bWaaSbaaSqaaiaaikdaaeqa aOGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGSaGaamiDamaaBa aaleaacaWGUbaabeaaaaa@40BB@  be a random sample of size n from exponential distribution. The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@  and the log likelihood function, lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gacaWGmbaaaa@38AB@ of exponential distribution are given by L= θ n e nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiabg2 da9iabeI7aXnaaCaaaleqabaGaamOBaaaakiaadwgadaahaaWcbeqa aiabgkHiTiaad6gacaaMc8UaeqiUdeNaaGPaVlqadshagaqeaaaaaa a@4381@  and lnL=nlnθnθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gacaWGmbGaeyypa0JaamOBaiGacYgacaGGUbGaeqiUdeNaeyOeI0Ia amOBaiaaykW7cqaH4oqCcaaMc8UabmiDayaaraaaaa@45FB@ . The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  of exponential distribution is the solution of the equation dlnL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaciiBaiaac6gacaWGmbaabaGaamizaiabeI7aXbaacqGH9aqp caaIWaaaaa@3E03@  and is given by θ ^ = 1 t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpdaWcaaqaaiaaigdaaeaaceWG0bGbaebaaaaaaa@3A9E@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaara aaaa@3707@ is the sample mean. Let t 1 , t 2 ,...., t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaaIXaaabeaakiaacYcacaWG0bWaaSbaaSqaaiaaikdaaeqa aOGaaiilaiaac6cacaGGUaGaaiOlaiaac6cacaGGSaGaamiDamaaBa aaleaacaWGUbaabeaaaaa@40BB@  be a random sample of size n from Lindley distribution. The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C7@  and the log likelihood function, lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gacaWGmbaaaa@38AB@ of Lindley distribution are given by L= ( θ 2 θ+1 ) n i=1 n ( 1+ t i ) e nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiabg2 da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaaa keaacqaH4oqCcqGHRaWkcaaIXaaaaaGaayjkaiaawMcaamaaCaaale qabaGaamOBaaaakmaarahabaWaaeWaaeaacaaIXaGaey4kaSIaamiD amaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawMcaaaWcbaGaamyAai abg2da9iaaigdaaeaacaWGUbaaniabg+GivdGccaaMc8Uaamyzamaa CaaaleqabaGaeyOeI0IaamOBaiaaykW7cqaH4oqCcaaMc8UabmiDay aaraaaaaaa@560B@  and lnL=nln( θ 2 θ+1 )+ i=1 n ln( 1+ t i ) nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gacaWGmbGaeyypa0JaamOBaiGacYgacaGGUbWaaeWaaeaadaWcaaqa aiabeI7aXnaaCaaaleqabaGaaGOmaaaaaOqaaiabeI7aXjabgUcaRi aaigdaaaaacaGLOaGaayzkaaGaey4kaSYaaabCaeaaciGGSbGaaiOB amaabmaabaGaaGymaiabgUcaRiaadshadaWgaaWcbaGaamyAaaqaba aakiaawIcacaGLPaaaaSqaaiaadMgacqGH9aqpcaaIXaaabaGaamOB aaqdcqGHris5aOGaeyOeI0IaamOBaiaaykW7cqaH4oqCcaaMc8Uabm iDayaaraaaaa@59D1@ . The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of the parameter θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of Lindley distribution is the solution of the equation dlnL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbGaciiBaiaac6gacaWGmbaabaGaamizaiabeI7aXbaacqGH9aqp caaIWaaaaa@3E03@  and is given by θ ^ = ( t ¯ 1 )+ ( t ¯ 1 ) 2 +8 t ¯ 2 t ¯ ; t ¯ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpdaWcaaqaaiabgkHiTmaabmaabaGabmiDayaaraGaeyOe I0IaaGymaaGaayjkaiaawMcaaiabgUcaRmaakaaabaWaaeWaaeaace WG0bGbaebacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaaGioaiaaykW7ceWG0bGbaebaaSqabaaake aacaaIYaGaaGPaVlqadshagaqeaaaacaGG7aGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UabmiDayaaraGaeyOpa4JaaGimaaaa@58A9@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaara aaaa@3707@ is the sample mean. It was shown by [5] showed that the estimator θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of Lindley distribution is positively biased, consistent and asymptotically normal.

From above table it is obvious that the fittings of Lindley distribution is better than the exponential distribution in Datasets 1-6,12,14,15. Whereas the fittings of exponential distribution is better than the Lindley distribution in Datasets 7-11,13 (Table 2).

Conclusion

In this paper we have tried to find the suitability of exponential and Lindley distributions for modeling real lifetimes data. It has been observed that neither exponential distribution nor Lindley distribution is appropriate for modeling real lifetime data in all cases. As per the nature of the data related with over-dispersion, equi-dispersion, and under-dispersion, in some cases exponential is better than Lindley while in other cases Lindley is better than exponential. Further, the decision about the suitability of exponential and Lindley for modeling real lifetime data depends on the nature of the data. Of course, Lindley is more flexible than exponential but exponential has some advantage over Lindley due to its simplicity.

Figure 1: Graphs of the p.d.f. of exponential and Lindley distributions (left hand side graphs are for exponential and right hand side graphs are for Lindley).
Figure 2: Graphs of the c.d.f. of exponential and Lindley distributions (left hand side graphs are for exponential and right hand side graphs are for Lindley).

Exponential Distribution

Lindley Distribution

C.V.= σ μ 1 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaac6 cacaWGwbGaaiOlaiabg2da9maalaaabaGaeq4WdmhabaGafqiVd0Mb auaadaWgaaWcbaGaaGymaaqabaaaaOGaeyypa0JaaGymaaaa@404A@

C.V.= σ μ 1 = θ 2 +4θ+2 θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaac6 cacaWGwbGaaiOlaiabg2da9maalaaabaGaeq4WdmhabaGafqiVd0Mb auaadaWgaaWcbaGaaGymaaqabaaaaOGaeyypa0ZaaSaaaeaadaGcaa qaaiabeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaisdacqaH 4oqCcqGHRaWkcaaIYaaaleqaaaGcbaGaeqiUdeNaey4kaSIaaGOmaa aaaaa@4AB5@

β 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaakiabg2da9iaaikdaaaa@3A5A@

β 1 = 2( θ 3 +6 θ 2 +6θ+2 ) ( θ 2 +4θ+2 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaakiabg2da9maalaaabaGa aGOmamaabmaabaGaeqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaS IaaGOnaiabeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiAda cqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaaabaWaaeWaaeaacq aH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI0aGaeqiUdeNa ey4kaSIaaGOmaaGaayjkaiaawMcaamaaCaaaleqabaWaaSGbaeaaca aIZaaabaGaaGOmaaaaaaaaaaaa@52C0@

β 2 =9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaOGaeyypa0JaaGyoaaaa@3A52@

β 2 = 3( 3 θ 4 +24 θ 3 +44 θ 2 +32θ+8 ) ( θ 2 +4θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIZaWaaeWaaeaa caaIZaGaeqiUde3aaWbaaSqabeaacaaI0aaaaOGaey4kaSIaaGOmai aaisdacqaH4oqCdaahaaWcbeqaaiaaiodaaaGccqGHRaWkcaaI0aGa aGinaiabeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiodaca aIYaGaeqiUdeNaey4kaSIaaGioaaGaayjkaiaawMcaaaqaamaabmaa baGaeqiUde3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinaiabeI 7aXjabgUcaRiaaikdaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikda aaaaaaaa@591E@

Index of dispersion
γ= σ 2 μ 1 = 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaSaaaeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakeaacqaH 8oqBdaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaOGamai4gkdiIc aaaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaeqiUdehaaaaa@44B6@

Index of dispersion
γ= σ 2 μ 1 = θ 2 +4θ+2 θ( θ 2 +3θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaSaaaeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakeaacqaH 8oqBdaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaOGamai4gkdiIc aaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcaaI0aGaeqiUdeNaey4kaSIaaGOmaaqaaiabeI7aXnaabm aabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaG4maiab eI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaaaaaaaa@54BD@

Table 1: Index of dispersion of exponential and Lindley distributions.

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

1.50

1.54

1.60

1.62

1.66

1.69

1.76

1.84

2.24

0.81

1.13

1.29

1.48

1.50

1.55

1.61

1.62

1.66

1.70

1.77

1.84

0.84

1.24

1.30

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 and Naylor [15].

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

83

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 and Saunders [16] 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 ( × MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqlaaa@380D@ 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaic dadaahaaWcbeqaaiabgkHiTiaaiodaaaaaaa@3942@ ) are presented below (after subtracting 65).

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

83

84

86

86

87

90

91

92

92

92

92

93

94

97

98

98

99

101

103

105

109

136

147

 

 

 

 

Data Set 3: The data set is from Lawless [17]. 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:

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 4: The data is from Picciotto [18] 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:

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 5: This data represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal [19].

12

15

22

24

24

32

32

33

34

38

38

43

44

48

52

53

54

54

55

56

57

58

58

59

60

60

60

60

61

62

63

65

65

67

68

70

70

72

73

75

76

76

81

83

84

85

87

91

95

96

98

99

109

110

121

127

129

131

143

146

146

175

175

211

233

258

258

263

297

341

341

376

 

 

 

 

 

 

Data Set 6: This data is related with behavioral sciences, collected by N Balakrishnan et al. [20]: 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:

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)

33

34

35(4)

36(2)

37(2)

39

42

44

Data Set 7: The data set reported by Efron [21] represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using radiotherapy (RT).  

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 8: The data set reported by Efron [21] 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).

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 9: This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee and Wang [22].

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 10: This data set is given by Linhart and Zucchini [23], which represents the failure times of the air conditioning system of an airplane:

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 11: This data set used by Bhaumik et al. [24], 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

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

23.0

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. [5] for fitting the Lindley [25] 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 [26]:

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 and Clark [27].

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. [28]:

 

Model

Parameter Estimate

-2ln L

AIC

AICC

BIC

K-S Statistic

 

 

 

 

 

 

 

 

Data 1

Lindley

0.996116

162.56

164.56

164.62

166.7

0.371

Exponential

0.663647

177.66

179.66

179.73

181.8

0.402

Data 2

Lindley

0.028859

983.11

985.11

985.15

987.71

0.242

Exponential

0.014635

1044.87

1046.87

1046.91

1049.48

0.357

Data 3

Lindley

0.027321

231.47

233.47

233.66

234.61

0.149

Exponential

0.013845

242.87

244.87

245.06

246.01

0.263

Data 4

Lindley

0.00897

1251.34

1253.34

1253.38

1255.95

0.098

Exponential

0.004505

1280.52

1282.52

1282.56

1285.12

0.19

Data 5

Lindley

0.019841

789.04

791.04

791.1

793.32

0.133

Exponential

0.010018

806.88

808.88

808.94

811.16

0.198

Data 6

Lindley

0.077247

1041.64

1043.64

1043.68

1046.54

0.448

Exponential

0.04006

1130.26

1132.26

1132.29

1135.16

0.525

Data 7

Lindley

0.008804

763.75

765.75

765.82

767.81

0.245

Exponential

0.004421

744.87

746.87

746.94

748.93

0.166

Data 8

Lindley

0.00891

579.16

581.16

581.26

582.95

0.219

Exponential

0.004475

564.02

566.02

566.11

567.8

0.145

Data 9

Lindley

0.196045

839.06

841.06

841.09

843.91

0.116

Exponential

0.106773

828.68

830.68

830.72

833.54

0.077

Data 10

Lindley

0.033021

323.27

325.27

325.42

326.67

0.345

Exponential

0.016779

305.26

307.26

307.4

308.66

0.213

Data 11

Lindley

0.823821

112.61

114.61

114.73

116.13

0.133

Exponential

0.532081

110.91

112.91

113.03

114.43

0.089

Data 12

Lindley

0.186571

638.07

640.07

640.12

642.68

0.058

Exponential

0.101245

658.04

660.04

660.08

662.65

0.163

Data 13

Lindley

0.01636

181.34

183.34

183.65

184.05

0.386

Exponential

0.008246

173.94

175.94

176.25

176.65

0.277

Data 14

Lindley

0.816118

60.5

62.5

62.72

63.49

0.341

Exponential

0.526316

65.67

67.67

67.9

68.67

0.389

Data 15

Lindley

0.062988

253.99

255.99

256.13

257.42

0.333

Exponential

0.032455

274.53

276.53

276.67

277.96

0.426

Table 2: MLE’s, -2ln L, AIC, AICC, BIC, K-S Statistics, and p-values of the fitted distributions of data sets 1-15.

References

  1. Davis DJ (1952) An analysis of some failure data. Journal of American Statistical Association 47(258): 113-150.
  2. Epstein B, Sobel M (1953) Life testing. Journal of American Statistical Association 48(263): 486-502.
  3. Epstein B (1958) The exponential distribution and its role in Life –testing. Ind Qual Control 15: 2-7.
  4. Feil P, Zelen M (1965) Estimation of exponential survival probabilities with concomitant information. Biometrics 21(4): 826-838.
  5. Ghitany ME, Atieh B, Nadarajah S (2008) Lidley distribution and its Applications. Mathematics Computing and Simulation 78(4): 493-506.
  6. Zakerzadah H, Dolati A (2010) Generalized Lindley distribution. Journal of Mathematics Extension 3(2): 13-25.
  7. Mazucheli J, Achcar JA (2011) The Lindley distribution applied to competing risks lifetime data. Comput Methods Programs Biomed 104(2): 188-192.
  8. Bakouch SH, Al-Zahrani BM, Al-Shomrani AA, Marchi VAA, Louzada F (2012) An extended Lindley distribution. Journal of Korean Statistical Society 41(1): 75-85.
  9. Shanker R, Mishra A (2013 a) A quasi Lindley distribution. African journal of Mathematics and Computer Science Research 6(4): 64-71.
  10.  Shanker R, Mishra A (2013 b) A two-parameter Lindley distribution. Statistics in transition new series 14(1): 45-56.
  11. Shanker R, Sharma S, Shanker R (2013) A two-parameter Lindley distribution for modeling waiting and survival times data. Applied Mathematics 4(2): 363-368.
  12. Shanker R, Amanuel AG (2013) A new quasi Lindley distribution. International Journal of Statistics and Systems 8(2): 143-156.
  13. Deniz EG, Ojeda EC (2011) The discrete Lindley distribution-Properties and Applications. Journal of Statistical Computation and Simulation 81(11): 1405-1416.
  14. Sankaran M (1970) The discrete Poisson-Lindley distribution. Biometrics 26(1): 145-149.
  15. Smith RL, Naylor JC (1987) A comparison of Maximum likelihood and Bayesian estimators for the three parameter Weibull distribution. Journal of the Royal Statistical Society 36(3): 358-369.
  16. Birnbaum ZW, Saunders SC (1969) Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability 6(2): 328-347.
  17. Lawless JF (1982) Statistical models and methods for lifetime data. John Wiley and Sons, New York, USA.
  18. Picciotto R (1970) Tensile fatigue characteristics of a sized polyester/viscose yarn and their effect on weaving performance, Master thesis, University of Raleigh, North Carolina State, USA.
  19. Bjerkedal T (1960) Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli. Am J Hyg 72: 130-148.
  20. 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.
  21. Efron B (1988) Logistic regression, survival analysis and the Kaplan-Meier curve. Journal of the American Statistical Association 83(402): 414-425.
  22. Lee ET, Wang JW (2003) Statistical methods for survival data analysis. (3rd edn), John Wiley and Sons, New York, USA.
  23. Linhart H, Zucchini W (1986) Model Selection, John Wiley, New York, USA.
  24. Bhaumik DK, Kapur K, Gibbons RD (2009) Testing Parameters of a Gamma Distribution for Small Samples. Technometrics 51(3): 326-334.
  25. Lindley DV (1958) Fiducial distributions and Bayes’ Theorem. Journal of the Royal Statistical Society 20(1): 102-107.
  26. Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 5(3): 375-383.
  27. Gross AJ, Clark VA (1975) Survival Distributions: Reliability Applications in the Biometrical Sciences, John Wiley, New York, USA.
  28. 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 Proceedings 419-430.
© 2014-2016 MedCrave Group, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use.
Creative Commons License Open Access by MedCrave Group is licensed under a Creative Commons Attribution 4.0 International License.
Based on a work at http://medcraveonline.com
Best viewed in Mozilla Firefox | Google Chrome | Above IE 7.0 version | Opera |Privacy Policy