ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 1 - 2016
On Two - Parameter Lindley Distribution and its Applications to Model Lifetime Data
Rama Shanker1*, Hagos Fesshaye2 and Shambhu Sharma3
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
3Department of Mathematics, Dayalbagh Educational Institute, India
Received:October 28, 2015 | Published: January 02, 2016
*Corresponding author: Ilker Etikan, Near East University, Nicosia-TRNC, Cyprus, Email:
Citation: Etikan I, Alkassim R, Abubakar S (2016) Comparision of Snowball Sampling and Sequential Sampling Technique. Biom Biostat Int J 3(1): 00056. DOI: 10.15406/bbij.2016.03.00056

Abstract

In this paper some of the important mathematical properties including moment generating function, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy and stress strength reliability of two-parameter Lindley distribution (TPLD) of Shanker & Mishra [1] have been discussed. Its goodness of fit over exponential and Lindley distributions have been illustrated with some real lifetime data-sets and found that TPLD is preferable over exponential and Lindley distributions for modeling lifetime data-sets.

Keywords: Mean deviations; Order statistics; Bonferroni and Lorenz curves; Entropy; Stress-strength reliability; Goodness of fit

Introduction

The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of distribution, introduced in the context of Bayesian analysis as a counter example of fiducial statistics, are given by

f( x;θ )= θ 2 θ+1 ( 1+x ) e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaW baaSqabeaajugWaiaaikdaaaaakeaajugibiabeI7aXjabgUcaRiaa igdaaaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiaadIhaaOGaay jkaiaawMcaaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbmacqGHsisl cqaH4oqCcaWG4baaaKqzGeGaaGPaVlaaykW7caaMc8UaaGPaVlaacU dacaWG4bGaeyOpa4JaaGimaiaacYcacaaMc8UaaGPaVlabeI7aXjab g6da+iaaicdaaaa@667C@     (1.1)

F( x;θ )=1[ 1+ θx θ+1 ] e θx ;x>0,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaH4oqCaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaaGymaiabgkHiTKqbaoaadmaakeaajugibi aaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacqaH4oqCcaWG4baakeaa jugibiabeI7aXjabgUcaRiaaigdaaaaakiaawUfacaGLDbaajugibi aadwgajuaGdaahaaWcbeqaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaa jugibiaaykW7caaMc8Uaai4oaiaadIhacqGH+aGpcaaIWaGaaiilai aaykW7cqaH4oqCcqGH+aGpcaaIWaaaaa@6127@     (1.2)

The detailed study about its mathematical properties, estimation of parameter and application showing the superiority of Lindley distribution over exponential distribution for the waiting times before service of the bank customers has been done by Ghitany et al. [2]. The Lindley distribution has been generalized extended and modified by different researchers including [1,3-19] are some among others.

The probability density function (p.d.f.) and cumulative distribution function (c.d.f) of two-parameter Lindley distribution (TPLD) of Shanker & Mishra [1] are given by

f( x;α,θ )= θ 2 αθ+1 ( α+x ) e θx ;x>0,θ>0,αθ>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaHXoqycaGGSaGaeqiU dehakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaakeaajugibi abeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaaGcbaqcLbsacqaH XoqycaaMc8UaeqiUdeNaey4kaSIaaGymaaaajuaGdaqadaGcbaqcLb sacqaHXoqycqGHRaWkcaWG4baakiaawIcacaGLPaaajugibiaadwga juaGdaahaaWcbeqaaKqzadGaeyOeI0IaeqiUdeNaamiEaaaajugibi aaykW7caaMc8UaaGPaVlaaykW7caGG7aGaamiEaiabg6da+iaaicda caGGSaGaaGPaVlaaykW7cqaH4oqCcqGH+aGpcaaIWaGaaiilaiabeg 7aHjaaykW7cqaH4oqCcqGH+aGpcqGHsislcaaIXaaaaa@7519@   (1.3)

F( x;α,θ )=1[ 1+αθ+θx αθ+1 ] e θx ;x>0,θ>0,αθ>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiEaiaacUdacqaHXoqycaGGSaGaeqiU dehakiaawIcacaGLPaaajugibiabg2da9iaaigdacqGHsisljuaGda WadaGcbaqcfa4aaSaaaOqaaKqzGeGaaGymaiabgUcaRiabeg7aHjaa ykW7cqaH4oqCcqGHRaWkcqaH4oqCcaWG4baakeaajugibiabeg7aHj aaykW7cqaH4oqCcqGHRaWkcaaIXaaaaaGccaGLBbGaayzxaaqcLbsa caWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiabeI7aXjaadIhaaa qcLbsacaGG7aGaamiEaiabg6da+iaaicdacaGGSaGaaGPaVlabeI7a Xjabg6da+iaaicdacaGGSaGaaGPaVlabeg7aHjaaykW7cqaH4oqCcq GH+aGpcqGHsislcaaIXaaaaa@7288@   (1.4)

At α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qycqGH9aqpcaaIXaaaaa@39E5@ , both (1.3) and (1.4) reduce to the corresponding expressions (1.1) and (1.2) of Lindley distribution. The first two moments about origin and the variance of TPLD of Shanker & Mishra [1] are given by

μ 1 = α θ + 2 θ ( α θ + 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9Kqbaoaalaaakeaajugibiabeg7aHj aaykW7cqaH4oqCcqGHRaWkcaaIYaaakeaajugibiabeI7aXLqbaoaa bmaakeaajugibiabeg7aHjaaykW7cqaH4oqCcqGHRaWkcaaIXaaaki aawIcacaGLPaaaaaaaaa@535C@   (1.5)

μ 2 = 2( αθ+3 ) θ 2 ( αθ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaaCaaaleqabaqc LbsacWaGGBOmGikaaiabg2da9Kqbaoaalaaakeaajugibiaaikdaju aGdaqadaGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaG4m aaGccaGLOaGaayzkaaaabaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaK qzadGaaGOmaaaajuaGdaqadaGcbaqcLbsacqaHXoqycaaMc8UaeqiU deNaey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa@596F@   (1.6)

μ 2 = α 2 θ 2 +4αθ+2 θ 2 ( αθ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaeqySdewcfa4aaWbaaSqabeaajugWaiaaikdaaa qcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiab gUcaRiaaisdacqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaGOmaaGcba qcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajuaGdaqa daGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaGymaaGcca GLOaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaaikdaaaaaaaaa@6061@   (1.7)

At α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qycqGH9aqpcaaIXaaaaa@39E5@ , these moments reduce to the corresponding moments of Lindley distribution. Shanker & Mishra [1] have derived and discussed some of its mathematical properties such as shape, moments, coefficient of variation, coefficient of skewness and kurtosis, hazard rate function, mean residual life function and stochastic orderings. They have also discussed the estimation of its parameters using maximum likelihood estimation and method of moments and its goodness of fit over Lindley distribution. It has been observed that many important mathematical properties of this distribution have not been studied.

In the present paper some of the important mathematical properties including moment generating function, mean deviations, order statistics, Bonferroni and Lorenz curves, Renyi entropy and stress strength reliability of TPLD of Shanker & Mishra [1] have been derived and discussed. Its goodness of fit over exponential and Lindley distributions have been illustrated with some real lifetime data-sets and found that TPLD gives better fit than exponential and Lindley distributions.

Moment Generating Function

The moment generating function, ( M X ( t ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamytaKqbaoaaBaaaleaajugibiaadIfaaSqabaqcfa4a aeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaacaGLOaGaayzkaa aaaa@3F5C@  of TPLD (1.3) can be obtained as

M X ( t )= θ 2 αθ+1 0 e ( θt ) ( α+x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb qcfa4aaSbaaSqaaKqzadGaamiwaaWcbeaajuaGdaqadaGcbaqcLbsa caWG0baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaakeaaju gibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaaGcbaqcLbsa cqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaGymaaaajuaGdaWdXbGcba qcLbsacaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTSWaaeWaaeaa jugWaiabeI7aXjabgkHiTiaadshaaSGaayjkaiaawMcaaaaajuaGda qadaGcbaqcLbsacqaHXoqycqGHRaWkcaWG4baakiaawIcacaGLPaaa aSqaaKqzadGaaGimaaWcbaqcLbmacqGHEisPaKqzGeGaey4kIipaca aMc8UaamizaiaadIhaaaa@688D@

= θ 2 αθ+1 [ α θt + 1 ( θt ) 2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGa aGOmaaaaaOqaaKqzGeGaeqySdeMaaGPaVlabeI7aXjabgUcaRiaaig daaaqcfa4aamWaaOqaaKqbaoaalaaakeaajugibiabeg7aHbGcbaqc LbsacqaH4oqCcqGHsislcaWG0baaaiabgUcaRKqbaoaalaaakeaaju gibiaaigdaaOqaaKqbaoaabmaakeaajugibiabeI7aXjabgkHiTiaa dshaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaa aaaOGaay5waiaaw2faaaaa@59AB@

= θ 2 αθ+1 [ α θ k=0 ( t θ ) k + 1 θ 2 k=0 ( k+1 k ) ( t θ ) k ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGa aGOmaaaaaOqaaKqzGeGaeqySdeMaaGPaVlabeI7aXjabgUcaRiaaig daaaqcfa4aamWaaOqaaKqbaoaalaaakeaajugibiabeg7aHbGcbaqc LbsacqaH4oqCaaqcfa4aaabCaOqaaKqbaoaabmaakeaajuaGdaWcaa GcbaqcLbsacaWG0baakeaajugibiabeI7aXbaaaOGaayjkaiaawMca aKqbaoaaCaaaleqabaqcLbmacaWGRbaaaKqzGeGaey4kaSscfa4aaS aaaOqaaKqzGeGaaGymaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqa aKqzadGaaGOmaaaaaaqcfa4aaabCaOqaaKqbaoaabmaakeaajugibu aabeqaceaaaOqaaKqzGeGaam4AaiabgUcaRiaaigdaaOqaaKqzGeGa am4AaaaaaOGaayjkaiaawMcaaKqbaoaabmaakeaajuaGdaWcaaGcba qcLbsacaWG0baakeaajugibiabeI7aXbaaaOGaayjkaiaawMcaaKqb aoaaCaaaleqabaqcLbmacaWGRbaaaaWcbaqcLbmacaWGRbGaeyypa0 JaaGimaaWcbaqcLbmacqGHEisPaKqzGeGaeyyeIuoaaSqaaKqzadGa am4Aaiabg2da9iaaicdaaSqaaKqzadGaeyOhIukajugibiabggHiLd aakiaawUfacaGLDbaaaaa@8287@

= k=0 ( αθ+1+k αθ+1 ) ( t θ ) k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaaeWbGcbaqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiab eg7aHjaaykW7cqaH4oqCcqGHRaWkcaaIXaGaey4kaSIaam4AaaGcba qcLbsacqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaGymaaaaaOGaayjk aiaawMcaaaWcbaqcLbmacaWGRbGaeyypa0JaaGimaaWcbaqcLbmacq GHEisPaKqzGeGaeyyeIuoajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqz GeGaamiDaaGcbaqcLbsacqaH4oqCaaaakiaawIcacaGLPaaajuaGda ahaaWcbeqaaKqzadGaam4Aaaaaaaa@5DB9@

 It can be easily seen that the expression for μ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaaaa@3FF6@ obtained as the coefficient of t r r! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamiDaKqbaoaaCaaaleqabaqcLbmacaWGYbaaaaGcbaqc LbsacaWGYbGaaiyiaaaaaaa@3D3B@  is given as

μ r = r!( αθ+r+1 ) θ r ( αθ+1 ) ;r=1,2,3,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaam OCaiaacgcajuaGdaqadaGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNa ey4kaSIaamOCaiabgUcaRiaaigdaaOGaayjkaiaawMcaaaqaaKqzGe GaeqiUdexcfa4aaWbaaSqabeaajugWaiaadkhaaaqcfa4aaeWaaOqa aKqzGeGaeqySdeMaaGPaVlabeI7aXjabgUcaRiaaigdaaOGaayjkai aawMcaaaaajugibiaaykW7caaMc8Uaai4oaiaadkhacqGH9aqpcaaI XaGaaiilaiaaikdacaGGSaGaaG4maiaacYcacaGGUaGaaiOlaiaac6 caaaa@6A85@

For α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qycqGH9aqpcaaIXaaaaa@39E5@ , μ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaaaa@3FF6@ reduces to the corresponding μ r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaWGYbaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaaaa@3FF6@ of Lindley distribution.

Mean Deviations

The amount of scatter in a population is measured to some extent by the totality of deviations usually from mean and median. These are known as the mean deviation about the mean and the mean deviation about the median defined by δ 1 ( X )= 0 | xμ | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aa8qCaOqaaK qbaoaaemaakeaajugibiaadIhacqGHsislcqaH8oqBaOGaay5bSlaa wIa7aaWcbaqcLbmacaaIWaaaleaajugWaiabg6HiLcqcLbsacqGHRi I8aiaaykW7caWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGa ayzkaaqcLbsacaWGKbGaamiEaaaa@58AE@  and δ 2 ( X )= 0 | xM | f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aa8qCaOqaaK qbaoaaemaakeaajugibiaadIhacqGHsislcaWGnbaakiaawEa7caGL iWoaaSqaaKqzadGaaGimaaWcbaqcLbmacqGHEisPaKqzGeGaey4kIi pacaaMc8UaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaa wMcaaKqzGeGaamizaiaadIhaaaa@57CB@ , respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBcqGH9aqpcaWGfbqcfa4aaeWaaOqaaKqzGeGaamiwaaGccaGLOaGa ayzkaaaaaa@3DA2@  and M=Median ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGnb Gaeyypa0JaaeytaiaabwgacaqGKbGaaeyAaiaabggacaqGUbGaaeii aKqbaoaabmaakeaajugibiaadIfaaOGaayjkaiaawMcaaaaa@41F7@ . The measures δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaaaa@3E6F@  and δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaaaa@3E70@ can be calculated using the relationships

δ 1 ( X )= 0 μ ( μx ) f( x )dx+ μ ( xμ ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aa8qCaOqaaK qbaoaabmaakeaajugibiabeY7aTjabgkHiTiaadIhaaOGaayjkaiaa wMcaaaWcbaqcLbmacaaIWaaaleaajugWaiabeY7aTbqcLbsacqGHRi I8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaa jugibiaadsgacaWG4bGaey4kaSscfa4aa8qCaOqaaKqbaoaabmaake aajugibiaadIhacqGHsislcqaH8oqBaOGaayjkaiaawMcaaaWcbaqc LbmacqaH8oqBaSqaaKqzadGaeyOhIukajugibiabgUIiYdGaamOzaK qbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaamiz aiaadIhaaaa@6D33@

=μF( μ ) 0 μ xf( x )dx μ[ 1F( μ ) ]+ μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqaH8oqBcaWGgbqcfa4aaeWaaOqaaKqzGeGaeqiVd0gakiaawIca caGLPaaajugibiabgkHiTKqbaoaapehakeaajugibiaadIhacaaMc8 UaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhaaSqaaKqzadGaaGimaaWcbaqcLbmacqaH8oqBaK qzGeGaey4kIipacqGHsislcqaH8oqBjuaGdaWadaGcbaqcLbsacaaI XaGaeyOeI0IaamOraKqbaoaabmaakeaajugibiabeY7aTbGccaGLOa GaayzkaaaacaGLBbGaayzxaaqcLbsacqGHRaWkjuaGdaWdXbGcbaqc LbsacaWG4bGaaGPaVdWcbaqcLbmacqaH8oqBaSqaaKqzadGaeyOhIu kajugibiabgUIiYdGaamOzaKqbaoaabmaakeaajugibiaadIhaaOGa ayjkaiaawMcaaKqzGeGaamizaiaadIhaaaa@73CF@

=2μF( μ )2μ+2 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIYaGaeqiVd0MaamOraKqbaoaabmaakeaajugibiabeY7aTbGc caGLOaGaayzkaaqcLbsacqGHsislcaaIYaGaeqiVd0Maey4kaSIaaG OmaKqbaoaapehakeaajugibiaadIhacaaMc8oaleaajugWaiabeY7a TbWcbaqcLbmacqGHEisPaKqzGeGaey4kIipacaWGMbqcfa4aaeWaaO qaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacaWGKbGaamiEaaaa @5803@

=2μF( μ )2 0 μ x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIYaGaeqiVd0MaamOraKqbaoaabmaakeaajugibiabeY7aTbGc caGLOaGaayzkaaqcLbsacqGHsislcaaIYaqcfa4aa8qCaOqaaKqzGe GaamiEaiaaykW7aSqaaKqzadGaaGimaaWcbaqcLbmacqaH8oqBaKqz GeGaey4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOa GaayzkaaqcLbsacaWGKbGaamiEaaaa@53F8@    (3.1)

and

δ 2 ( X )= 0 M ( Mx ) f( x )dx+ M ( xM ) f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aa8qCaOqaaK qbaoaabmaakeaajugibiaad2eacqGHsislcaWG4baakiaawIcacaGL PaaaaSqaaKqzadGaaGimaaWcbaqcLbmacaWGnbaajugibiabgUIiYd GaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhacqGHRaWkjuaGdaWdXbGcbaqcfa4aaeWaaOqaaK qzGeGaamiEaiabgkHiTiaad2eaaOGaayjkaiaawMcaaaWcbaqcLbma caWGnbaaleaajugWaiabg6HiLcqcLbsacqGHRiI8aiaadAgajuaGda qadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaajugibiaadsgacaWG 4baaaa@69A4@

=MF( M ) 0 M xf( x )dx M[ 1F( M ) ]+ M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaWGnbGaaGPaVlaadAeajuaGdaqadaGcbaqcLbsacaWGnbaakiaa wIcacaGLPaaajugibiabgkHiTKqbaoaapehakeaajugibiaadIhaca aMc8UaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMca aKqzGeGaamizaiaadIhaaSqaaKqzadGaaGimaaWcbaqcLbmacaWGnb aajugibiabgUIiYdGaeyOeI0IaamytaKqbaoaadmaakeaajugibiaa igdacqGHsislcaWGgbqcfa4aaeWaaOqaaKqzGeGaamytaaGccaGLOa GaayzkaaaacaGLBbGaayzxaaqcLbsacqGHRaWkjuaGdaWdXbGcbaqc LbsacaWG4bGaaGPaVdWcbaqcLbmacaWGnbaaleaajugWaiabg6HiLc qcLbsacqGHRiI8aiaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaa wIcacaGLPaaajugibiaadsgacaWG4baaaa@7002@

=μ+2 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqGHsislcqaH8oqBcqGHRaWkcaaIYaqcfa4aa8qCaOqaaKqzGeGa amiEaiaaykW7aSqaaKqzadGaamytaaWcbaqcLbmacqGHEisPaKqzGe Gaey4kIipacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGa ayzkaaqcLbsacaWGKbGaamiEaaaa@4E27@

=μ2 0 M x f( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcqaH8oqBcqGHsislcaaIYaqcfa4aa8qCaOqaaKqzGeGaamiEaiaa ykW7aSqaaKqzadGaaGimaaWcbaqcLbmacaWGnbaajugibiabgUIiYd GaamOzaKqbaoaabmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqz GeGaamizaiaadIhaaaa@4C8E@    (3.2)

Using p.d.f. (1.3), and expression for mean of two-parameter Lindley distribution, we have

0 μ xf( x ) dx=μ { θ 2 ( μ 2 +αμ )+2θμ+( αθ+2 ) } e θμ θ( αθ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaiaaykW7caWGMbqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaaaleaajugWaiaaicdaaSqaaKqzadGaeqiVd0 gajugibiabgUIiYdGaaGPaVlaadsgacaWG4bGaeyypa0JaeqiVd0Ma eyOeI0scfa4aaSaaaOqaaKqbaoaacmaakeaajugibiabeI7aXLqbao aaCaaaleqabaqcLbmacaaIYaaaaKqbaoaabmaakeaajugibiabeY7a TLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaeqySde MaaGPaVlabeY7aTbGccaGLOaGaayzkaaqcLbsacqGHRaWkcaaIYaGa eqiUdeNaaGPaVlabeY7aTjabgUcaRKqbaoaabmaakeaajugibiabeg 7aHjaaykW7cqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaaaiaa wUhacaGL9baajugibiaadwgajuaGdaahaaWcbeqaaKqzadGaeyOeI0 IaeqiUdeNaaGPaVlabeY7aTbaaaOqaaKqzGeGaeqiUdexcfa4aaeWa aOqaaKqzGeGaeqySdeMaaGPaVlabeI7aXjabgUcaRiaaigdaaOGaay jkaiaawMcaaaaaaaa@88EE@     (3.3)

Using expressions from (3.1), (3.2) and (3.3), and little algebraic simplification, the mean deviation about mean, δ 1 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaaaa@3E6F@ and the mean deviation about median, δ 2 ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaaaa@3E70@ of TPLD (1.3) are obtained as

δ 1 ( X )= 2( θμ+αθ+2 ) e θμ θ( αθ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qzGeGaaGOmaKqbaoaabmaakeaajugibiabeI7aXjaaykW7cqaH8oqB cqGHRaWkcqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaGOmaaGccaGLOa GaayzkaaqcLbsacaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiab eI7aXjaaykW7cqaH8oqBaaaakeaajugibiabeI7aXLqbaoaabmaake aajugibiabeg7aHjaaykW7cqaH4oqCcqGHRaWkcaaIXaaakiaawIca caGLPaaaaaaaaa@65D4@     (3.4)

and δ 2 ( X )= 2{ θ 2 ( M 2 +αM )+2θM+( αθ+2 ) } e θM θ( αθ+1 ) μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH0o azjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaabmaakeaajugi biaadIfaaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qzGeGaaGOmaKqbaoaacmaakeaajugibiabeI7aXLqbaoaaCaaaleqa baqcLbmacaaIYaaaaKqbaoaabmaakeaajugibiaad2eajuaGdaahaa WcbeqaaKqzadGaaGOmaaaajugibiabgUcaRiabeg7aHjaaykW7caWG nbaakiaawIcacaGLPaaajugibiabgUcaRiaaikdacqaH4oqCcaaMc8 UaamytaiabgUcaRKqbaoaabmaakeaajugibiabeg7aHjaaykW7cqaH 4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPaaaaiaawUhacaGL9baaju gibiaadwgajuaGdaahaaWcbeqaaKqzadGaeyOeI0IaeqiUdeNaaGPa Vlaad2eaaaaakeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeg 7aHjaaykW7cqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaaaaqc LbsacqGHsislcqaH8oqBaaa@7CBD@     (3.5)

It can be easily seen that expressions (3.4) and (3.5) of TPLD (1.3) reduce to the corresponding expressions of Lindley distribution at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0JaaGymaaaa@3956@ .

Order Statistics

Let X 1 , X 2 ,..., X n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaacYcacaaMc8Ua amiwaKqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacaGGSaGaaG PaVlaac6cacaGGUaGaaiOlaiaacYcacaaMc8UaamiwaKqbaoaaBaaa leaajugWaiaad6gaaSqabaaaaa@4B44@  be a random sample of size  from two-parameter Lindley distribution (1.3). Let X ( 1 ) < X ( 2 ) <...< X ( n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb WcdaWgaaqaamaabmaabaqcLbmacaaIXaaaliaawIcacaGLPaaaaeqa aKqzGeGaeyipaWJaamiwaSWaaSbaaeaadaqadaqaaKqzadGaaGOmaa WccaGLOaGaayzkaaaabeaajugibiabgYda8iaaykW7caaMc8UaaiOl aiaac6cacaGGUaGaaGPaVlaaykW7cqGH8aapcaWGybWcdaWgaaqaam aabmaabaqcLbmacaWGUbaaliaawIcacaGLPaaaaeqaaaaa@50BC@ denote the corresponding order statistics. The p.d.f. and the c.d.f. of the k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ th order statistic, say Y= X ( k ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb Gaeyypa0JaamiwaSWaaSbaaeaadaqadaqaaKqzadGaam4AaaWccaGL OaGaayzkaaaabeaaaaa@3D24@ are given by

f Y ( y )= n! ( k1 )!( nk )! F k1 ( y ) { 1F( y ) } nk f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzadGaamywaaWcbeaajuaGdaqadaGcbaqcLbsa caWG5baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaakeaaju gibiaad6gacaGGHaaakeaajuaGdaqadaGcbaqcLbsacaWGRbGaeyOe I0IaaGymaaGccaGLOaGaayzkaaqcLbsacaGGHaGaaGPaVNqbaoaabm aakeaajugibiaad6gacqGHsislcaWGRbaakiaawIcacaGLPaaajugi biaacgcaaaGaaGPaVlaadAeajuaGdaahaaWcbeqaaKqzadGaam4Aai abgkHiTiaaigdaaaqcfa4aaeWaaOqaaKqzGeGaamyEaaGccaGLOaGa ayzkaaqcfa4aaiWaaOqaaKqzGeGaaGymaiabgkHiTiaadAeajuaGda qadaGcbaqcLbsacaWG5baakiaawIcacaGLPaaaaiaawUhacaGL9baa juaGdaahaaWcbeqaaKqzadGaamOBaiabgkHiTiaadUgaaaqcLbsaca WGMbqcfa4aaeWaaOqaaKqzGeGaamyEaaGccaGLOaGaayzkaaaaaa@6F33@

= n! ( k1 )!( nk )! l=0 nk ( nk l ) ( 1 ) l F k+l1 ( y )f( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWcaaGcbaqcLbsacaWGUbGaaiyiaaGcbaqcfa4aaeWaaOqa aKqzGeGaam4AaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaai yiaiaaykW7juaGdaqadaGcbaqcLbsacaWGUbGaeyOeI0Iaam4AaaGc caGLOaGaayzkaaqcLbsacaGGHaaaaiaaykW7juaGdaaeWbGcbaqcfa 4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsacaWGUbGaeyOeI0Ia am4AaaGcbaqcLbsacaWGSbaaaaGccaGLOaGaayzkaaaaleaajugWai aadYgacqGH9aqpcaaIWaaaleaajugWaiaad6gacqGHsislcaWGRbaa jugibiabggHiLdqcfa4aaeWaaOqaaKqzGeGaeyOeI0IaaGymaaGcca GLOaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaadYgaaaqcLbsacaWG gbqcfa4aaWbaaSqabeaajugWaiaadUgacqGHRaWkcaWGSbGaeyOeI0 IaaGymaaaajuaGdaqadaGcbaqcLbsacaWG5baakiaawIcacaGLPaaa jugibiaadAgajuaGdaqadaGcbaqcLbsacaWG5baakiaawIcacaGLPa aaaaa@7590@

and

F Y ( y )= j=k n ( n j ) F j ( y ) { 1F( y ) } nj MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzadGaamywaaWcbeaajuaGdaqadaGcbaqcLbsa caWG5baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaaqahakeaaju aGdaqadaGcbaqcLbsafaqabeGabaaakeaajugibiaad6gaaOqaaKqz GeGaamOAaaaaaOGaayjkaiaawMcaaaWcbaqcLbmacaWGQbGaeyypa0 Jaam4AaaWcbaqcLbmacaWGUbaajugibiabggHiLdGaaGPaVlaadAea juaGdaahaaWcbeqaaKqzadGaamOAaaaajuaGdaqadaGcbaqcLbsaca WG5baakiaawIcacaGLPaaajuaGdaGadaGcbaqcLbsacaaIXaGaeyOe I0IaamOraKqbaoaabmaakeaajugibiaadMhaaOGaayjkaiaawMcaaa Gaay5Eaiaaw2haaKqbaoaaCaaaleqabaqcLbmacaWGUbGaeyOeI0Ia amOAaaaaaaa@6605@

= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l F j+l ( y ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaaeWbGcbaqcfa4aaabCaOqaaKqbaoaabmaakeaajugibuaa beqaceaaaOqaaKqzGeGaamOBaaGcbaqcLbsacaWGQbaaaaGccaGLOa GaayzkaaaaleaajugWaiaadYgacqGH9aqpcaaIWaaaleaajugWaiaa d6gacqGHsislcaWGQbaajugibiabggHiLdqcfa4aaeWaaOqaaKqzGe qbaeqabiqaaaGcbaqcLbsacaWGUbGaeyOeI0IaamOAaaGcbaqcLbsa caWGSbaaaaGccaGLOaGaayzkaaaaleaajugWaiaadQgacqGH9aqpca WGRbaaleaajugWaiaad6gaaKqzGeGaeyyeIuoacaaMc8Ecfa4aaeWa aOqaaKqzGeGaeyOeI0IaaGymaaGccaGLOaGaayzkaaqcfa4aaWbaaS qabeaajugWaiaadYgaaaqcLbsacaWGgbqcfa4aaWbaaSqabeaajugW aiaadQgacqGHRaWkcaWGSbaaaKqbaoaabmaakeaajugibiaadMhaaO GaayjkaiaawMcaaaaa@6C81@

respectively, for k=1,2,3,...,n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb Gaeyypa0JaaGymaiaacYcacaaIYaGaaiilaiaaiodacaGGSaGaaiOl aiaac6cacaGGUaGaaiilaiaad6gaaaa@4078@

Thus, the p.d.f. and the c.d.f of the th order statistics of TPLD (1.3) are obtained as

f Y ( y )= n! θ 2 ( α+x ) e θx ( αθ+1 )( k1 )!( nk )! l=0 nk ( nk l ) ( 1 ) l × [ 1 1+αθ+θx αθ+1 e θx ] k+l1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzadGaamywaaWcbeaajuaGdaqadaGcbaqcLbsa caWG5baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaalaaakeaaju gibiaad6gacaGGHaGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikda aaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaey4kaSIaamiEaaGccaGLOa GaayzkaaqcLbsacaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiab eI7aXjaadIhaaaaakeaajuaGdaqadaGcbaqcLbsacqaHXoqycaaMc8 UaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaeWaaOqa aKqzGeGaam4AaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqzGeGaai yiaiaaykW7juaGdaqadaGcbaqcLbsacaWGUbGaeyOeI0Iaam4AaaGc caGLOaGaayzkaaqcLbsacaGGHaaaaiaaykW7juaGdaaeWbGcbaqcfa 4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsacaWGUbGaeyOeI0Ia am4AaaGcbaqcLbsacaWGSbaaaaGccaGLOaGaayzkaaaaleaajugWai aadYgacqGH9aqpcaaIWaaaleaajugWaiaad6gacqGHsislcaWGRbaa jugibiabggHiLdqcfa4aaeWaaOqaaKqzGeGaeyOeI0IaaGymaaGcca GLOaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaadYgaaaqcLbsacqGH xdaTjuaGdaWadaGcbaqcLbsacaaIXaGaeyOeI0scfa4aaSaaaOqaaK qzGeGaaGymaiabgUcaRiabeg7aHjaaykW7cqaH4oqCcqGHRaWkcqaH 4oqCcaaMb8UaamiEaaGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNaey 4kaSIaaGymaaaacaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiab eI7aXjaadIhaaaaakiaawUfacaGLDbaajuaGdaahaaWcbeqaaKqzad Gaam4AaiabgUcaRiaadYgacqGHsislcaaIXaaaaaaa@AFE8@

and

F Y ( y )= j=k n l=0 nj ( n j ) ( nj l ) ( 1 ) l [ 1 1+αθ+θx αθ+1 e θx ] j+l MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzadGaamywaaWcbeaajuaGdaqadaGcbaqcLbsa caWG5baakiaawIcacaGLPaaajugibiabg2da9Kqbaoaaqahakeaaju aGdaaeWbGcbaqcfa4aaeWaaOqaaKqzGeqbaeqabiqaaaGcbaqcLbsa caWGUbaakeaajugibiaadQgaaaaakiaawIcacaGLPaaaaSqaaKqzad GaamiBaiabg2da9iaaicdaaSqaaKqzadGaamOBaiabgkHiTiaadQga aKqzGeGaeyyeIuoajuaGdaqadaGcbaqcLbsafaqabeGabaaakeaaju gibiaad6gacqGHsislcaWGQbaakeaajugibiaadYgaaaaakiaawIca caGLPaaaaSqaaKqzadGaamOAaiabg2da9iaadUgaaSqaaKqzadGaam OBaaqcLbsacqGHris5aiaaykW7juaGdaqadaGcbaqcLbsacqGHsisl caaIXaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzadGaamiBaa aajuaGdaWadaGcbaqcLbsacaaIXaGaeyOeI0scfa4aaSaaaOqaaKqz GeGaaGymaiabgUcaRiabeg7aHjaaykW7cqaH4oqCcqGHRaWkcqaH4o qCcaaMb8UaamiEaaGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNaey4k aSIaaGymaaaacaWGLbqcfa4aaWbaaSqabeaajugibiabgkHiTiabeI 7aXjaadIhaaaaakiaawUfacaGLDbaajuaGdaahaaWcbeqaaKqzadGa amOAaiabgUcaRiaadYgaaaaaaa@8DDA@

It can be easily verified that the expressions for the p.d.f. and c.d.f. of the th order statistics of TPLD (1.3) reduce to the expressions for the p.d.f. and c.d.f. of the th order statistics of Lindley distribution at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qycqGH9aqpcaaIXaaaaa@39E5@

Bonferroni and Lorenz Curves

The Bonferroni and Lorenz curves [20] and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

B( p )= 1 pμ 0 q xf( x ) dx= 1 pμ [ 0 xf( x )dx q xf( x ) dx ]= 1 pμ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadchacqaH8o qBaaqcfa4aa8qCaOqaaKqzGeGaamiEaiaaykW7caWGMbqcfa4aaeWa aOqaaKqzGeGaamiEaaGccaGLOaGaayzkaaqcLbsacaaMc8oaleaaju gWaiaaicdaaSqaaKqzadGaamyCaaqcLbsacqGHRiI8aiaadsgacaWG 4bGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaWGWb GaeqiVd0gaaKqbaoaadmaakeaajuaGdaWdXbGcbaqcLbsacaWG4bGa aGPaVlaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPa aajugibiaadsgacaWG4bGaeyOeI0caleaajugWaiaaicdaaSqaaKqz adGaeyOhIukajugibiabgUIiYdqcfa4aa8qCaOqaaKqzGeGaamiEai aaykW7caWGMbqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzk aaaaleaajugWaiaadghaaSqaaKqzadGaeyOhIukajugibiabgUIiYd GaaGPaVlaadsgacaWG4baakiaawUfacaGLDbaajugibiabg2da9Kqb aoaalaaakeaajugibiaaigdaaOqaaKqzGeGaamiCaiabeY7aTbaaju aGdaWadaGcbaqcLbsacqaH8oqBcqGHsisljuaGdaWdXbGcbaqcLbsa caWG4bGaaGPaVlaadAgajuaGdaqadaGcbaqcLbsacaWG4baakiaawI cacaGLPaaaaSqaaKqzadGaamyCaaWcbaqcLbmacqGHEisPaKqzGeGa ey4kIipacaaMc8UaamizaiaadIhaaOGaay5waiaaw2faaaaa@A4A0@ (5.1)

L( p )= 1 μ 0 q xf( x ) dx= 1 μ [ 0 xf( x )dx q xf( x ) dx ]= 1 μ [ μ q xf( x ) dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiabeY7aTbaaju aGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaadAgajuaGdaqadaGcbaqc LbsacaWG4baakiaawIcacaGLPaaaaSqaaKqzadGaaGimaaWcbaqcLb macaWGXbaajugibiabgUIiYdGaaGPaVlaadsgacaWG4bGaeyypa0tc fa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacqaH8oqBaaqcfa4aam WaaOqaaKqbaoaapehakeaajugibiaadIhacaaMc8UaamOzaKqbaoaa bmaakeaajugibiaadIhaaOGaayjkaiaawMcaaKqzGeGaamizaiaadI hacqGHsislaSqaaKqzadGaaGimaaWcbaqcLbmacqGHEisPaKqzGeGa ey4kIipajuaGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaadAgajuaGda qadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaajugibiaaykW7aSqa aKqzadGaamyCaaWcbaqcLbmacqGHEisPaKqzGeGaey4kIipacaWGKb GaamiEaaGccaGLBbGaayzxaaqcLbsacqGH9aqpjuaGdaWcaaGcbaqc LbsacaaIXaaakeaajugibiabeY7aTbaajuaGdaWadaGcbaqcLbsacq aH8oqBcqGHsisljuaGdaWdXbGcbaqcLbsacaWG4bGaaGPaVlaadAga juaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGLPaaaaSqaaKqzad GaamyCaaWcbaqcLbmacqGHEisPaKqzGeGaey4kIipacaaMc8Uaamiz aiaadIhaaOGaay5waiaaw2faaaaa@A1CB@ (5.2)

respectively or equivalently

B( p )= 1 pμ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadchacqaH8o qBaaqcfa4aa8qCaOqaaKqzGeGaamOraKqbaoaaCaaaleqabaqcLbma cqGHsislcaaIXaaaaKqbaoaabmaakeaajugibiaadIhaaOGaayjkai aawMcaaaWcbaqcLbmacaaIWaaaleaajugWaiaadchaaKqzGeGaey4k IipacaaMc8UaamizaiaadIhaaaa@5572@ (5.3)

and L( p )= 1 μ 0 p F 1 ( x ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiabeY7aTbaaju aGdaWdXbGcbaqcLbsacaWGgbqcfa4aaWbaaSqabeaajugWaiabgkHi Tiaaigdaaaqcfa4aaeWaaOqaaKqzGeGaamiEaaGccaGLOaGaayzkaa aaleaajugWaiaaicdaaSqaaKqzadGaamiCaaqcLbsacqGHRiI8aiaa ykW7caWGKbGaamiEaaaa@5487@ (5.4)

respectively, where μ=E( X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBcqGH9aqpcaWGfbqcfa4aaeWaaOqaaKqzGeGaamiwaaGccaGLOaGa ayzkaaaaaa@3DA2@ and q= F 1 ( p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGXb Gaeyypa0JaamOraKqbaoaaCaaaleqabaqcLbmacqGHsislcaaIXaaa aKqbaoaabmaakeaajugibiaadchaaOGaayjkaiaawMcaaaaa@408C@ .

The Bonferroni and Gini indices are thus defined as

B=1 0 1 B( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb Gaeyypa0JaaGymaiabgkHiTKqbaoaapehakeaajugibiaadkeajuaG daqadaGcbaqcLbsacaWGWbaakiaawIcacaGLPaaaaSqaaKqzadGaaG imaaWcbaqcLbmacaaIXaaajugibiabgUIiYdGaaGPaVlaadsgacaWG Wbaaaa@49C7@ (5.5)

and G=12 0 1 L( p ) dp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaaGymaiabgkHiTiaaikdajuaGdaWdXbGcbaqcLbsacaWG mbqcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsaca aMc8oaleaajugWaiaaicdaaSqaaKqzadGaaGymaaqcLbsacqGHRiI8 aiaadsgacaWGWbaaaa@4B21@ (5.6)

respectively.

Using p.d.f. (1.3), we get

q xf( x ) dx= { θ 2 ( q 2 +αq )+2θq+( αθ+2 ) } e θq θ( αθ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aa8qCaO qaaKqzGeGaamiEaiaaykW7caWGMbqcfa4aaeWaaOqaaKqzGeGaamiE aaGccaGLOaGaayzkaaaaleaajugWaiaadghaaSqaaKqzadGaeyOhIu kajugibiabgUIiYdGaaGPaVlaadsgacaWG4bGaeyypa0tcfa4aaSaa aOqaaKqbaoaacmaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLb macaaIYaaaaKqbaoaabmaakeaajugibiaadghajuaGdaahaaWcbeqa aKqzadGaaGOmaaaajugibiabgUcaRiabeg7aHjaaykW7caWGXbaaki aawIcacaGLPaaajugibiabgUcaRiaaikdacqaH4oqCcaWGXbGaey4k aSscfa4aaeWaaOqaaKqzGeGaeqySdeMaaGPaVlabeI7aXjabgUcaRi aaikdaaOGaayjkaiaawMcaaaGaay5Eaiaaw2haaKqzGeGaamyzaKqb aoaaCaaaleqabaqcLbmacqGHsislcqaH4oqCcaWGXbaaaaGcbaqcLb sacqaH4oqCjuaGdaqadaGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNa ey4kaSIaaGymaaGccaGLOaGaayzkaaaaaaaa@802C@ (5.7)

Now using equation (5.7) in (5.1) and (5.2), we get

B( p )= 1 p [ 1 { θ 2 ( q 2 +αq )+2θq+( αθ+2 ) } e θq αθ+2 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaadchaaaqcfa 4aamWaaOqaaKqzGeGaaGymaiabgkHiTKqbaoaalaaakeaajuaGdaGa daGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaaju aGdaqadaGcbaqcLbsacaWGXbqcfa4aaWbaaSqabeaajugWaiaaikda aaqcLbsacqGHRaWkcqaHXoqycaaMc8UaamyCaaGccaGLOaGaayzkaa qcLbsacqGHRaWkcaaIYaGaeqiUdeNaamyCaiabgUcaRKqbaoaabmaa keaajugibiabeg7aHjaaykW7cqaH4oqCcqGHRaWkcaaIYaaakiaawI cacaGLPaaaaiaawUhacaGL9baajugibiaadwgajuaGdaahaaWcbeqa aKqzadGaeyOeI0IaeqiUdeNaamyCaaaaaOqaaKqzGeGaeqySdeMaaG PaVlabeI7aXjabgUcaRiaaikdaaaaakiaawUfacaGLDbaaaaa@7621@ (5.8)

and L( p )=1 { θ 2 ( q 2 +αq )+2θq+( αθ+2 ) } e θq αθ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamiCaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqbaoaacmaakeaajugibi abeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqbaoaabmaakeaa jugibiaadghajuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiabgU caRiabeg7aHjaaykW7caWGXbaakiaawIcacaGLPaaajugibiabgUca RiaaikdacqaH4oqCcaWGXbGaey4kaSscfa4aaeWaaOqaaKqzGeGaeq ySdeMaaGPaVlabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaGa ay5Eaiaaw2haaKqzGeGaamyzaKqbaoaaCaaaleqabaqcLbmacqGHsi slcqaH4oqCcaWGXbaaaaGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNa ey4kaSIaaGOmaaaaaaa@6F88@ (5.9)

Now using equations (5.8) and (5.9) in (5.5) and (5.6), the Bonferroni and Gini indices of TPLD (1.3) are obtained as

B=1 { θ 2 ( q 2 +αq )+2θq+( αθ+2 ) } e θq αθ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb Gaeyypa0JaaGymaiabgkHiTKqbaoaalaaakeaajuaGdaGadaGcbaqc LbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajuaGdaqada GcbaqcLbsacaWGXbqcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsa cqGHRaWkcqaHXoqycaaMc8UaamyCaaGccaGLOaGaayzkaaqcLbsacq GHRaWkcaaIYaGaeqiUdeNaamyCaiabgUcaRKqbaoaabmaakeaajugi biabeg7aHjaaykW7cqaH4oqCcqGHRaWkcaaIYaaakiaawIcacaGLPa aaaiaawUhacaGL9baajugibiaadwgajuaGdaahaaWcbeqaaKqzadGa eyOeI0IaeqiUdeNaamyCaaaaaOqaaKqzGeGaeqySdeMaaGPaVlabeI 7aXjabgUcaRiaaikdaaaaaaa@6B40@ (5.10)

G=1+ 2{ θ 2 ( q 2 +αq )+2θq+( αθ+2 ) } e θq αθ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGhb Gaeyypa0JaeyOeI0IaaGymaiabgUcaRKqbaoaalaaakeaajugibiaa ikdajuaGdaGadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaGOmaaaajuaGdaqadaGcbaqcLbsacaWGXbqcfa4aaWbaaSqabeaa jugWaiaaikdaaaqcLbsacqGHRaWkcqaHXoqycaaMc8UaamyCaaGcca GLOaGaayzkaaqcLbsacqGHRaWkcaaIYaGaeqiUdeNaamyCaiabgUca RKqbaoaabmaakeaajugibiabeg7aHjaaykW7cqaH4oqCcqGHRaWkca aIYaaakiaawIcacaGLPaaaaiaawUhacaGL9baajugibiaadwgajuaG daahaaWcbeqaaKqzadGaeyOeI0IaeqiUdeNaamyCaaaaaOqaaKqzGe GaeqySdeMaaGPaVlabeI7aXjabgUcaRiaaikdaaaaaaa@6D72@ (5.11)

The Bonferroni and Gini indices of Lindley distribution are particular cases of the Bonferroni and Gini indices (5.10) and (5.11) of TPLD (1.3) for α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qycqGH9aqpcaaIXaaaaa@39E5@ .

Renyi Entropy

An entropy of a random variable is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy [21]. If is a continuous random variable having probability density function f( . ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaaiOlaaGccaGLOaGaayzkaaaaaa@3ADC@ , then Renyi entropy is defined as

T R ( γ )= 1 1γ log{ f γ ( x )dx } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub qcfa4aaSbaaSqaaKqzadGaamOuaaWcbeaajuaGdaqadaGcbaqcLbsa cqaHZoWzaOGaayjkaiaawMcaaKqzGeGaeyypa0tcfa4aaSaaaOqaaK qzGeGaaGymaaGcbaqcLbsacaaIXaGaeyOeI0Iaeq4SdCgaaiGacYga caGGVbGaai4zaKqbaoaacmaakeaajuaGdaWdbaGcbaqcLbsacaWGMb qcfa4aaWbaaSqabeaajugWaiabeo7aNbaajuaGdaqadaGcbaqcLbsa caWG4baakiaawIcacaGLPaaajugibiaadsgacaWG4baaleqabeqcLb sacqGHRiI8aaGccaGL7bGaayzFaaaaaa@59E6@

where γ>0andγ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH+aGpcaaIWaGaaGPaVlaaykW7caaMc8Uaaeyyaiaab6gacaqG KbGaaGPaVlaaykW7caaMc8Uaeq4SdCMaeyiyIKRaaGymaaaa@4A15@ .

Thus, the Renyi entropy for TPLD (1.3) can be obtained as

The Renyi entropy of Lindley distribution is a particular case of the Renyi entropy TPLD at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qycqGH9aqpcaaIXaaaaa@39E5@ .

Stress-Strength Reliability

The stress- strength reliability describes the life of a component which has random strength that is subjected to a random stress . When the stress applied to it exceeds the strength, the component fails instantly and the component will function satisfactorily till X>Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb GaeyOpa4Jaamywaaaa@3948@ . Therefore, R=P( Y<X ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb Gaeyypa0JaamiuaKqbaoaabmaakeaajugibiaadMfacqGH8aapcaWG ybaakiaawIcacaGLPaaaaaa@3EB0@ is a measure of component reliability and in statistical literature it is known as stress-strength parameter. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.

Let X MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb aaaa@3762@ and Y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGzb aaaa@3763@ be independent strength and stress random variables having TPLD (1.3) with parameter ( α 1 , θ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqySdewcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugi biaacYcacqaH4oqCjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaaGcca GLOaGaayzkaaaaaa@42A0@ and ( α 2 , θ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqySdewcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugi biaacYcacqaH4oqCjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaaGcca GLOaGaayzkaaaaaa@42A2@ respectively. Then the stress-strength reliability R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb aaaa@375C@ is obtained as

R=P( Y<X )= 0 P( Y<X|X=x ) f X ( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGsb Gaeyypa0JaamiuaKqbaoaabmaakeaajugibiaadMfacqGH8aapcaWG ybaakiaawIcacaGLPaaajugibiabg2da9Kqbaoaapehakeaajugibi aadcfajuaGdaqadaGcbaqcLbsacaWGzbGaeyipaWJaamiwaiaacYha caWGybGaeyypa0JaamiEaaGccaGLOaGaayzkaaaaleaajugWaiaaic daaSqaaKqzadGaeyOhIukajugibiabgUIiYdGaamOzaKqbaoaaBaaa leaajugWaiaadIfaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaaGcca GLOaGaayzkaaqcLbsacaWGKbGaamiEaaaa@5CFE@

= 0 f( x; α 1 , θ 1 ) F( x; α 2 , θ 2 )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpjuaGdaWdXbGcbaqcLbsacaWGMbqcfa4aaeWaaOqaaKqzGeGaamiE aiaacUdacqaHXoqyjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGe GaaiilaiabeI7aXLqbaoaaBaaaleaajugWaiaaigdaaSqabaaakiaa wIcacaGLPaaaaSqaaKqzadGaaGimaaWcbaqcLbmacqGHEisPaKqzGe Gaey4kIipacaaMc8UaaGPaVlaadAeajuaGdaqadaGcbaqcLbsacaWG 4bGaai4oaiabeg7aHLqbaoaaBaaaleaajugWaiaaikdaaSqabaqcLb sacaGGSaGaeqiUdexcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaOGa ayjkaiaawMcaaKqzGeGaamizaiaadIhaaaa@643E@

=1 θ 1 2 [ 2 θ 2 +( α 1 θ 2 + α 2 θ 2 +1 )( θ 1 + θ 2 )+ α 1 ( α 2 θ 2 +1 ) ( θ 1 + θ 2 ) 2 ] ( α 1 θ 1 +1 )( α 2 θ 2 +1 ) ( θ 1 + θ 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGH9a qpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaSba aSqaaKqzadGaaGymaaWcbeaajuaGdaahaaWcbeqaaKqzadGaaGOmaa aajuaGdaWadaGcbaqcLbsacaaIYaGaeqiUdexcfa4aaSbaaSqaaKqz adGaaGOmaaWcbeaajugibiabgUcaRKqbaoaabmaakeaajugibiabeg 7aHLqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsacqaH4oqCjuaG daWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaey4kaSIaeqySdewcfa 4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabeI7aXLqbaoaaBaaa leaajugWaiaaikdaaSqabaqcLbsacqGHRaWkcaaIXaaakiaawIcaca GLPaaajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaWgaaWcbaqcLbma caaIXaaaleqaaKqzGeGaey4kaSIaeqiUdexcfa4aaSbaaSqaaKqzad GaaGOmaaWcbeaaaOGaayjkaiaawMcaaKqzGeGaey4kaSIaeqySdewc fa4aaSbaaSqaaKqzadGaaGymaaWcbeaajuaGdaqadaGcbaqcLbsacq aHXoqyjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeqiUdexc fa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabgUcaRiaaigdaaO GaayjkaiaawMcaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaBaaa leaajugWaiaaigdaaSqabaqcLbsacqGHRaWkcqaH4oqCjuaGdaWgaa WcbaqcLbmacaaIYaaaleqaaaGccaGLOaGaayzkaaqcfa4aaWbaaSqa beaajugWaiaaikdaaaaakiaawUfacaGLDbaaaeaajuaGdaqadaGcba qcLbsacqaHXoqyjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGa eqiUdexcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiabgUcaRi aaigdaaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiabeg7aHLqb aoaaBaaaleaajugWaiaaikdaaSqabaqcLbsacqaH4oqCjuaGdaWgaa WcbaqcLbmacaaIYaaaleqaaKqzGeGaey4kaSIaaGymaaGccaGLOaGa ayzkaaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaSbaaSqaaKqzad GaaGymaaWcbeaajugibiabgUcaRiabeI7aXLqbaoaaBaaaleaajugW aiaaikdaaSqabaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzad GaaGOmaaaaaaaaaa@C05C@

The expression of stress-strength reliability of Lindley distribution is a particular case of the expression of stress-strength reliability of TPLD (1.3) at α 1 = α 2 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qyjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqzGeGaeyypa0JaeqyS dewcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiabg2da9iaaig daaaa@4305@ .

Estimation of Parameters

  1. Method of moment estimate of parameters

The TPLD (1.3) has two parameters to be estimated and so the first two moments about origin are required to estimate parameters. Using the first two moments about origin, we have

μ 2 ( μ 1 ) 2 =k(Say)= 2( αθ+3 )( αθ+1 ) ( αθ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH8oqBdaWgaaqcfasaaiaaikdaaKqbagqaamaaCaaabeqaaiad acUHYaIOaaaabaWaaeWaaeaacqaH8oqBdaWgaaqcfasaaiaaigdaaK qbagqaamaaCaaabeqaaiadacUHYaIOaaaacaGLOaGaayzkaaWaaWba aeqajuaibaGaaGOmaaaaaaqcfaOaeyypa0Jaam4AaiaacIcacaqGtb GaaeyyaiaabMhacaGGPaGaeyypa0ZaaSaaaeaacaaIYaWaaeWaaeaa cqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaG4maaGaayjkaiaawMcaam aabmaabaGaeqySdeMaaGPaVlabeI7aXjabgUcaRiaaigdaaiaawIca caGLPaaaaeaadaqadaqaaiabeg7aHjaaykW7cqaH4oqCcqGHRaWkca aIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaaaaaaaa@6760@ (8.1.1)

Taking b=αθ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGIb Gaeyypa0JaeqySdeMaaGPaVlabeI7aXbaa@3D52@ , we get

μ 2 ( μ 1 ) 2 = 2( b+3 )( b+1 ) ( b+2 ) 2 = 2 b 2 +8b+6 b 2 +4b+4 =k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH8oqBdaWgaaqcfasaaiaaikdaaKqbagqaamaaCaaabeqaaiad acUHYaIOaaaabaWaaeWaaeaacqaH8oqBdaWgaaqcfasaaiaaigdaaK qbagqaamaaCaaabeqaaiadacUHYaIOaaaacaGLOaGaayzkaaWaaWba aeqajuaibaGaaGOmaaaaaaqcfaOaeyypa0ZaaSaaaeaacaaIYaWaae WaaeaacaWGIbGaey4kaSIaaG4maaGaayjkaiaawMcaamaabmaabaGa amOyaiabgUcaRiaaigdaaiaawIcacaGLPaaaaeaadaqadaqaaiaadk gacqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOm aaaaaaqcfaOaeyypa0ZaaSaaaeaacaaIYaGaamOyamaaCaaabeqcfa saaiaaikdaaaqcfaOaey4kaSIaaGioaiaadkgacqGHRaWkcaaI2aaa baGaamOyamaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGinai aadkgacqGHRaWkcaaI0aaaaiabg2da9iaadUgaaaa@671C@

This gives a quadratic equation in b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGIb aaaa@376C@ as

( 2k ) b 2 +4( 2k )b+2( 32k )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaaGOmaiabgkHiTiaadUgaaOGaayjkaiaawMcaaKqzGeGa amOyaKqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaG inaKqbaoaabmaakeaajugibiaaikdacqGHsislcaWGRbaakiaawIca caGLPaaajugibiaadkgacqGHRaWkcaaIYaqcfa4aaeWaaOqaaKqzGe GaaG4maiabgkHiTiaaikdacaWGRbaakiaawIcacaGLPaaajugibiab g2da9iaaicdaaaa@5259@ (8.1.2)

Replacing the first and second moments about origin μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaaaa@3FBA@ and μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaaaa@3FBB@ by their respective sample moments, an estimate of k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ can be obtained and substituting the value of k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ in equation (8.1.2), an estimate of can be obtained. Substituting this estimate of in the expression for the mean of TPLD (1.3), moment estimate θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaacaaaa@384A@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ can be obtained as

θ ˜ =( b+2 b+1 ) 1 x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaacaiabg2da9KqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWG IbGaey4kaSIaaGOmaaGcbaqcLbsacaWGIbGaey4kaSIaaGymaaaaaO GaayjkaiaawMcaaKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa bmiEayaaraaaaaaa@45F4@ (8.1.3)

Finally, moment estimate α ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaHXo qygaacaaaa@3833@ of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qyaaa@3824@ can be obtained as

θ ˜ =( b+2 b+1 ) 1 x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaacaiabg2da9KqbaoaabmaakeaajuaGdaWcaaGcbaqcLbsacaWG IbGaey4kaSIaaGOmaaGcbaqcLbsacaWGIbGaey4kaSIaaGymaaaaaO GaayjkaiaawMcaaKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa bmiEayaaraaaaaaa@45F4@ (8.1.3)

Finally, moment estimate α ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaHXo qygaacaaaa@3833@ of α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qyaaa@3824@ can be obtained as

α ˜ = b θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaHXo qygaacaiabg2da9KqbaoaalaaakeaajugibiaadkgaaOqaaKqzGeGa fqiUdeNbaGaaaaaaaa@3DB5@

b. Maximum likelihood estimate of parameters

Let ( x 1 , x 2 , x 3 ,..., x n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaamiEaKqbaoaaBaaaleaajugWaiaaigdaaSqabaqcLbsa caGGSaGaaGPaVlaadIhajuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaK qzGeGaaiilaiaaykW7caWG4bqcfa4aaSbaaSqaaKqzadGaaG4maaWc beaajugibiaacYcacaaMc8UaaGPaVlaac6cacaGGUaGaaiOlaiaayk W7caaMc8UaaiilaiaadIhajuaGdaWgaaWcbaqcLbmacaWGUbaaleqa aaGccaGLOaGaayzkaaaaaa@575C@ be a random sample from TPLD (1.3). Let f x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaSbaaSqaaKqzadGaamiEaaWcbeaaaaa@3A60@ be the observed frequency in the sample corresponding to X=x(x=1,2,3,...k) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGyb Gaeyypa0JaamiEaiaaykW7caGGOaGaamiEaiabg2da9iaaigdacaGG SaGaaGOmaiaacYcacaaIZaGaaiilaiaac6cacaGGUaGaaiOlaiaadU gacaGGPaaaaa@4596@ such that x=1 k f x =n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaabCaO qaaKqzGeGaamOzaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaabaqc LbmacaWG4bGaeyypa0JaaGymaaWcbaqcLbmacaWGRbaajugibiabgg HiLdGaeyypa0JaamOBaaaa@45BF@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ is the largest observed value having non-zero frequency. The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb aaaa@3756@ of TPLD (1.3) is given by

L= ( θ 2 αθ+1 ) n i=1 n ( α+x ) f x e nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0tcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiabeI7aXLqb aoaaCaaaleqabaqcLbmacaaIYaaaaaGcbaqcLbsacqaHXoqycaaMc8 UaeqiUdeNaey4kaSIaaGymaaaaaOGaayjkaiaawMcaaKqbaoaaCaaa leqabaqcLbmacaWGUbaaaKqbaoaarahakeaajuaGdaqadaGcbaqcLb sacqaHXoqycqGHRaWkcaWG4baakiaawIcacaGLPaaajuaGdaahaaWc beqaaKqzadGaamOzaSWaaSbaaWqaaKqzadGaamiEaaadbeaaaaaale aajugWaiaadMgacqGH9aqpcaaIXaaaleaajugWaiaad6gaaKqzGeGa ey4dIunacaaMc8UaamyzaKqbaoaaCaaaleqabaqcLbmacqGHsislca WGUbGaaGPaVlabeI7aXjaaykW7ceWG4bGbaebaaaaaaa@6AF9@ (8.2.1)

The log likelihood function is thus obtained as

logL=nlog θ 2 nlog( αθ+1 )+ x=1 k f x log( α+x ) nθ x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb Gaai4BaiaacEgacaWGmbGaeyypa0JaamOBaiGacYgacaGGVbGaai4z aiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaeyOeI0 IaamOBaiGacYgacaGGVbGaai4zaKqbaoaabmaakeaajugibiabeg7a HjaaykW7cqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajugibi abgUcaRKqbaoaaqahakeaajugibiaadAgajuaGdaWgaaWcbaqcLbma caWG4baaleqaaKqzGeGaciiBaiaac+gacaGGNbqcfa4aaeWaaOqaaK qzGeGaeqySdeMaey4kaSIaamiEaaGccaGLOaGaayzkaaaaleaajugW aiaadIhacqGH9aqpcaaIXaaaleaajugWaiaadUgaaKqzGeGaeyyeIu oacqGHsislcaWGUbGaaGPaVlabeI7aXjaaykW7ceWG4bGbaebaaaa@7264@ (8.2.2)

where x ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b Gbaebaaaa@379A@ is the sample mean.

The two log likelihood equations are obtained as

logL θ = 2n θ nα αθ+1 n x ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaciiBaiaac+gacaGGNbGaamitaaGcbaqcLbsa cqGHciITcqaH4oqCaaGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmai aad6gaaOqaaKqzGeGaeqiUdehaaiabgkHiTKqbaoaalaaakeaajugi biaad6gacqaHXoqyaOqaaKqzGeGaeqySdeMaaGPaVlabeI7aXjabgU caRiaaigdaaaGaeyOeI0IaamOBaiqadIhagaqeaiabg2da9iaaicda aaa@56A5@

logL α = nθ αθ+1 + x=1 k f x α+x =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyRaciiBaiaac+gacaGGNbGaamitaaGcbaqcLbsa cqGHciITcqaHXoqyaaGaeyypa0JaeyOeI0scfa4aaSaaaOqaaKqzGe GaamOBaiabeI7aXbGcbaqcLbsacqaHXoqycaaMc8UaeqiUdeNaey4k aSIaaGymaaaacqGHRaWkjuaGdaaeWbGcbaqcfa4aaSaaaOqaaKqzGe GaamOzaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaakeaajugibiab eg7aHjabgUcaRiaadIhaaaaaleaajugWaiaadIhacaaMc8Uaeyypa0 JaaGymaaWcbaqcLbmacaWGRbaajugibiabggHiLdGaeyypa0JaaGim aaaa@6382@

It can be easily seen that equation (8.2.3) gives x ¯ = αθ+2 θ( αθ+1 ) = μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG4b GbaebacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHXoqycaaMc8UaeqiU deNaey4kaSIaaGOmaaGcbaqcLbsacqaH4oqCjuaGdaqadaGcbaqcLb sacqaHXoqycaaMc8UaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzk aaaaaKqzGeGaeyypa0JaeqiVd0wcfa4aaSbaaSqaaKqzadGaaGymaa WcbeaajuaGdaahaaWcbeqaaKqzadGamai4gkdiIcaaaaa@56A5@ , mean of TPLD. The equations (8.2.3) and (8.2.4) do not seem to be solved directly. However, Fisher’s scoring method can be applied to solve these equations iteratively. We have

2 logL θ 2 = 2n θ 2 + n α 2 ( αθ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyBcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsa ciGGSbGaai4BaiaacEgacaWGmbaakeaajugibiabgkGi2kabeI7aXL qbaoaaCaaaleqabaqcLbmacaaIYaaaaaaajugibiabg2da9iabgkHi TKqbaoaalaaakeaajugibiaaikdacaWGUbaakeaajugibiabeI7aXL qbaoaaCaaaleqabaqcLbmacaaIYaaaaaaajugibiabgUcaRKqbaoaa laaakeaajugibiaad6gacqaHXoqyjuaGdaahaaWcbeqaaKqzadGaaG OmaaaaaOqaaKqbaoaabmaakeaajugibiabeg7aHjaaykW7cqaH4oqC cqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzad GaaGOmaaaaaaaaaa@63E3@

2 logL θα = n ( αθ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyBcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsa ciGGSbGaai4BaiaacEgacaWGmbaakeaajugibiabgkGi2kabeI7aXj aaykW7cqGHciITcqaHXoqyaaGaeyypa0JaeyOeI0scfa4aaSaaaOqa aKqzGeGaamOBaaGcbaqcfa4aaeWaaOqaaKqzGeGaeqySdeMaaGPaVl abeI7aXjabgUcaRiaaigdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqa baqcLbmacaaIYaaaaaaaaaa@57B0@ (8.2.6)

2 logL α 2 = n θ 2 ( αθ+1 ) 2 x=1 k f x ( α+x ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeyOaIyBcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsa ciGGSbGaai4BaiaacEgacaWGmbaakeaajugibiabgkGi2kabeg7aHL qbaoaaCaaaleqabaqcLbmacaaIYaaaaaaajugibiabg2da9Kqbaoaa laaakeaajugibiaad6gacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG OmaaaaaOqaaKqbaoaabmaakeaajugibiabeg7aHjaaykW7cqaH4oqC cqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzad GaaGOmaaaaaaqcLbsacqGHsisljuaGdaaeWbGcbaqcfa4aaSaaaOqa aKqzGeGaamOzaKqbaoaaBaaaleaajugWaiaadIhaaSqabaaakeaaju aGdaqadaGcbaqcLbsacqaHXoqycqGHRaWkcaWG4baakiaawIcacaGL PaaajuaGdaahaaWcbeqaaKqzadGaaGOmaaaaaaaaleaajugWaiaadI hacqGH9aqpcaaIXaaaleaajugWaiaadUgaaKqzGeGaeyyeIuoaaaa@7291@ (8.2.7)

The maximum likelihood estimates of parameters are the solution of the following equations

[ 2 logL θ 2 2 logL θα 2 logL θα 2 logL α 2 ] θ ^ = θ 0 α ^ = α 0 [ θ ^ = θ 0 α ^ = α 0 ]= [ logL θ logL α ] θ ^ = θ 0 α ^ = α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aamWaaO qaaKqzGeqbaeqabiGaaaGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyBc fa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsaciGGSbGaai4BaiaacE gacaWGmbaakeaajugibiabgkGi2kabeI7aXLqbaoaaCaaaleqabaqc LbmacaaIYaaaaaaaaOqaaKqbaoaalaaakeaajugibiabgkGi2Mqbao aaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaciiBaiaac+gacaGGNbGa amitaaGcbaqcLbsacqGHciITcqaH4oqCcaaMc8UaeyOaIyRaeqySde gaaaGcbaqcfa4aaSaaaOqaaKqzGeGaeyOaIyBcfa4aaWbaaSqabeaa jugWaiaaikdaaaqcLbsaciGGSbGaai4BaiaacEgacaWGmbaakeaaju gibiabgkGi2kabeI7aXjaaykW7cqGHciITcqaHXoqyaaaakeaajuaG daWcaaGcbaqcLbsacqGHciITjuaGdaahaaWcbeqaaKqzadGaaGOmaa aajugibiGacYgacaGGVbGaai4zaiaadYeaaOqaaKqzGeGaeyOaIyRa eqySdewcfa4aaWbaaSqabeaajugWaiaaikdaaaaaaaaaaOGaay5wai aaw2faaKqbaoaaBaaaleaajugibuaabeqaceaaaSqaaKqzadGafqiU deNbaKaacqGH9aqpcqaH4oqClmaaBaaameaajugWaiaaicdaaWqaba aaleaajugWaiqbeg7aHzaajaGaeyypa0JaeqySde2cdaWgaaadbaqc LbmacaaIWaaameqaaaaaaSqabaqcfa4aamWaaOqaaKqzGeqbaeqabi qaaaGcbaqcLbsacuaH4oqCgaqcaiabg2da9iabeI7aXLqbaoaaBaaa leaajugibiaaicdaaSqabaaakeaajugibiqbeg7aHzaajaGaeyypa0 JaeqySdewcfa4aaSbaaSqaaKqzGeGaaGimaaWcbeaaaaaakiaawUfa caGLDbaajugibiabg2da9KqbaoaadmaakeaajugibuaabeqaceaaaO qaaKqbaoaalaaakeaajugibiabgkGi2kGacYgacaGGVbGaai4zaiaa dYeaaOqaaKqzGeGaeyOaIyRaeqiUdehaaaGcbaqcfa4aaSaaaOqaaK qzGeGaeyOaIyRaciiBaiaac+gacaGGNbGaamitaaGcbaqcLbsacqGH ciITcqaHXoqyaaaaaaGccaGLBbGaayzxaaqcfa4aaSbaaSqaaKqzGe qbaeqabiqaaaWcbaqcLbmacuaH4oqCgaqcaiabg2da9iabeI7aXTWa aSbaaWqaaKqzadGaaGimaaadbeaaaSqaaKqzadGafqySdeMbaKaacq GH9aqpcqaHXoqylmaaBaaameaajugWaiaaicdaaWqabaaaaaWcbeaa aaa@C83E@

where θ 0 and α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCjuaGdaWgaaWcbaqcLbmacaaIWaaaleqaaKqzGeGaaGPaVlaabgga caqGUbGaaeizaiaaykW7cqaHXoqyjuaGdaWgaaWcbaqcLbmacaaIWa aaleqaaaaa@4595@ are initial values of θandα MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcaaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlabeg7aHbaa@3FAC@ as given by the method of moments. These equations are solved iteratively till sufficiently close estimates of θ ^ and α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaiaaykW7caaMc8Uaaeyyaiaab6gacaqGKbGaaGPaVlqbeg7a Hzaajaaaaa@4157@ are obtained.

Applications of Two-Parameter Lindley Distribution

The two-parameter Lindley distribution (TPLD) has been fitted to a number of lifetime data- sets. In this section, we present the fitting of two-parameter Lindley distribution to five real lifetime data-sets and compare its goodness of fit with the one parameter exponential and Lindley distributions data sets (1-5).

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 1: This data set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross et al. [22].

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 2: This data set is the strength data of glass of the aircraft window reported by Fuller et al. [23].

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 3: This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al. [2] for fitting the Lindley [24] distribution.

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 4: 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 [25]

17.88

28.92

33

41.52

42.12

45.6

48.8

51.84

51.96

54.12

55.56

67.8

68.44

68.64

68.88

84.12

93.12

98.64

105.12

105.84

127.92

128.04

173.4

Data set 5: The data set is from Lawless [26]. 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:

In order to compare distributions, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics (Kolmogorov-Smirnov Statistics) for five real data - sets have been computed (Table 1). The formulae for computing AIC, AICC, BIC, and K-S Statistics are as follows:

 

Model

Estimate of Parameters

— 2ln L

AIC

AICC

BIC

K-S
Statistics

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

α ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqySdeMbaK aaaaa@37A5@

Data 1

Lindley

0.816118

 

60.50

62.50

62.72

63.49

0.341

 

Exponential
TPLD

0.526316
1.545110

 

— 0.31285

65.67
40.71

67.67
44.71

67.90
45.41

68.67
46.70

0.389
0.204

Data 2

Lindley

0.062988

 

253.99

255.99

256.13

257.42

0.333

 

Exponential
TPLD

0.032455
0.103985

 

— 5.25330

274.53
231.82

276.53
235.82

276.67
236.25

277.96
238.69

0.426
0.298

Data 3

Lindley

0.186571

 

638.07

640.07

640.12

642.68

0.058

 

Exponential
TPLD

0.101245
0.196210

 

0.337078

658.04
635.75

660.04
639.75

660.08
639.87

662.65
639.75

0.163
0.040

Data 4

Lindley

0.996116

 

162.56

164.56

164.62

166.70

0.371

 

Exponential
TPLD

0.663647
2.146474

 

0.257373

177.66
91.56

179.66
95.56

179.73
95.63

181.80
97.36

0.402
0.361

Data 5

Lindley

0.027321

 

231.47

233.47

233.66

234.61

0.149

 

Exponential
TPLD

0.013845
0.035434

 

10.12355

242.87
223.52

244.87
227.52

245.06
228.12

246.01
229.79

0.263
0.098

Table 1: MLE’s, — 2ln L, AIC, AICC, BIC, K-S Statistics of the fitted distributions of data sets 1-5.

AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaamysaiaadoeacqGH9aqpcqGHsislcaaIYaGaciiBaiaac6gacaWG mbGaey4kaSIaaGOmaiaadUgaaaa@40D3@ ,

AICC=AIC+ 2k( k+1 ) ( nk1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaamysaiaadoeacaWGdbGaeyypa0JaamyqaiaadMeacaWGdbGaey4k aSscfa4aaSaaaOqaaKqzGeGaaGOmaiaadUgajuaGdaqadaGcbaqcLb sacaWGRbGaey4kaSIaaGymaaGccaGLOaGaayzkaaaabaqcfa4aaeWa aOqaaKqzGeGaamOBaiabgkHiTiaadUgacqGHsislcaaIXaaakiaawI cacaGLPaaaaaaaaa@4D49@ ,

BIC=2lnL+klnn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb GaamysaiaadoeacqGH9aqpcqGHsislcaaIYaGaciiBaiaac6gacaWG mbGaey4kaSIaam4AaiGacYgacaGGUbGaaGPaVlaad6gaaaa@447A@ and

D= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaeyypa0tcfa4aaCbeaOqaaKqzGeGaae4uaiaabwhacaqGWbaaleaa jugWaiaadIhaaSqabaqcfa4aaqWaaOqaaKqzGeGaamOraKqbaoaaBa aaleaajugWaiaad6gaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaaGc caGLOaGaayzkaaqcLbsacqGHsislcaWGgbqcfa4aaSbaaSqaaKqzad GaaGimaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGL PaaaaiaawEa7caGLiWoaaaa@5307@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ = the number of parameters, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@ = the sample size and F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaSbaaSqaaKqzadGaamOBaaWcbeaajuaGdaqadaGcbaqcLbsa caWG4baakiaawIcacaGLPaaaaaa@3DED@ is the empirical distribution function.

The best distribution corresponds to lower 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC, AICC, BIC, and K-S statistics.

Conclusion

In the present paper some of the important mathematical properties including moment generating function, mean deviations, order statistics, Bonferroni and Lorenz curves, entropy and stress strength reliability of two-parameter Lindley distribution (TPLD) of Shanker & Mishra [1] have been derived and discussed. The distribution has been fitted to some real lifetime data-sets to test its goodness of fit over exponential and Lindley distributions. It is obvious from the fitting of TPLD that it gives better fitting than exponential and Lindley distributions and hence TPLD is preferable over exponential and Lindley distributions for modeling lifetime data-sets from different fields of knowledge.

References

  1.  Shanker R, Mishra A (2013 a) A two-parameter Lindley distribution. Statistics in Transition-new series 14(1): 45-56.
  2. Ghitany ME, Atieh B, Nadarajah S (2008) Lidley distribution and its Application. Mathematics Computing and Simulation 78(4): 493-506.
  3. Zakerzadeh H, Dolati A (2009) Generalized Lindley distribution. Journal of Mathematical extension 3(2): 13-25.
  4. Nadarajah S, Bakouch HS, Tahmasbi R (2011) A generalized Lindley distribution. Sankhya Series 73(2): 331-359.
  5. Deniz E, Ojeda E (2011) The discrete Lindley distribution-Properties and Applications. Journal of Statistical Computation and Simulation 81(11): 1405-1416.
  6. Bakouch HS, Al Zaharani B, Al Shomrani A, Marchi V, and Louzad F (2012) An extended Lindley distribution. Journal of the Korean Statistical Society 41(1): 75-85.
  7. Shanker R, Mishra A (2013 b) A quasi Lindley distribution. African Journal of Mathematics and Computer Science Research 6(4): 64-71.
  8. 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.
  9. Elbatal I, Merovi F, Elgarhy M (2013) A new generalized Lindley distribution. Mathematical theory and Modeling 3(13): 30-47.
  10. Ghitany M, Al Mutairi D, Balakrishnan N, Al Enezi I (2013) Power Lindley distribution and associated inference. Computational Statistics and Data Analysis 64: 20-33.
  11. Merovci F (2013) Transmuted Lindley distribution. International Journal of Open Problems in Computer Science and Mathematics 6(2): 63-72.
  12. Liyanage GW, Pararai M (2014) A generalized Power Lindley distribution with applications. Asian journal of Mathematics and Applications 1-23.
  13. Ashour S, Eltehiwy M (2014) Exponentiated Power Lindley distribution. Journal of Advanced Research 6(6): 895-905.
  14. Oluyede BO, Yang T (2014) A new class of generalized Lindley distribution with applications. Journal of Statistical Computation and Simulation 85(10): 2072-2100.
  15. Singh SK, Singh U, Sharma VK (2014) The Truncated Lindley distribution-inference and Application. Journal of Statistics Applications & Probability 3(2): 219-228.
  16. Sharma V, Singh S, Singh U, Agiwal V (2015) The inverse Lindley distribution-A stress-strength reliability model with applications to head and neck cancer data. Journal of Industrial &Production Engineering 32(3): 162-173.
  17. 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.
  18. Alkarni S (2015) Extended Power Lindley distribution-A new Statistical model for non-monotone survival data. European journal of statistics and probability 3(3): 19-34.
  19. Pararai M, Liyanage GW, Oluyede BO (2015) A new class of generalized Power Lindley distribution with applications to lifetime data.Theoretical Mathematics &Applications 5(1): 53-96.
  20. Bonferroni CE (1930) Elementi di Statistca generale. Seeber, Firenze, Itlay.
  21. Renyi A (1961) On measures of entropy and information in proceedings of the 4th berkeley symposium on Mathematical Statistics and Probability. University of California press, Berkeley, USA, 1: 547-561.
  22. Gross AJ, Clark VA (1975) Survival Distributions Reliability Applications in the Biometrical Sciences. John Wiley, New York, USA.
  23. 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.
  24. Lindley DV (1958) Fiducial distributions and Bayes’ theorem. Journal of the Royal Statistical Society. SeriesB 20(1): 102-107.
  25. 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.
  26. Lawless JF (1982) Statistical models and methods for lifetime data. John Wiley and Sons, New York, USA.
© 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