eISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Creative Review
Volume 7 Issue 3

A retrospective study on Lindley distribution

Lishamol Tomy
Department of Statistics, Deva Matha College, India
Received: February 15, 2018 | Published: April 03, 2018

Correspondence: Lishamol Tomy, Department of Statistics, Deva Matha College, Kuravilangad, Kerala, 686633, India, Email

Citation: Tomy L. A retrospective study on Lindley distribution. Biom Biostat Int J. 2018;7(3):163‒169. DOI: 10.15406/bbij.2018.07.00205

Abstract

Generalizations make a distribution more flexible especially for studying the tail properties. There exist many generalized family of continuous univariate distributions. In this article, a survey on Lindley distribution, its extensions and classes is conducted. Several available generalizations of the distribution are reviewed and recent trends in the construction of generalized classes are discussed.

Keywords: generalizations, lindley distribution, power distribution, truncated distributions, wrapped distributions

Introduction

The analysis and modeling of lifetime data are very important in applied sciences such as engineering, public health, actuarial science, biomedical studies, demography, industrial reliability and other applied sciences. Therefore, it seems crucial to find statistical distributions for model real‒world phenomena. There are a number of lifetime distributions in statistical literature including exponential, Weibull, gamma and lognormal distributions. The Weibull distribution is one of the most popular and widely used models in life testing and reliability theory.1 Introduced a one‒parameter distribution known as Lindley distribution. This model is used as an alternative model for existing statistical distributions. Lindley distribution has several real applications where the data show the non‒monotone shape for their hazard rate. Generalizations of existing distribution is an another interest in statistical research. The extended distributions have attracted in statistical literature to develop new models. The transformations of distributions have been proved useful in exploring skewness and tail properties, and also for improving the goodness‒of‒fit of the extended family. The Lindley distribution has been generalized by many authors in recent years. This study concentrates on conducting an extensive enquiry on those different fields of existing knowledge.

This paper is organized as follows. Section 2 deals with the Lindley Distribution. Section 3 discusses existing generalizations of Lindley distribution. Finally, concluding remarks are given in Section 4.

Lindley distribution

The probability density function (pdf) of one parameter Lindley distribution is given by

f(x;θ)= θ 2 1+θ (1+x) e -θx ; x>0, θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWFMb Gaa8hkaiaa=HhacaWF7aaccaGae4hUdeNaa8xkaiaa=1dadaWcaaqa aiab+H7aXnaaCaaaleqabaGaa8NmaaaaaOqaaiaa=fdacaWFRaGae4 hUdehaaiaa=HcacaWFXaGaa83kaiaa=HhacaWFPaGaa8xzamaaCaaa leqabaGaa8xlaiab+H7aXjaa=HhaaaGccaWFGaGaa83oaiaa=bcaca WFGaGaa8hEaiaa=5dacaWFWaGaa8hlaiaa=bcacqGF4oqCcaWF+aGa a8hmaaaa@562C@ (2.1)

The cumulative density function (cdf) of one parameter Lindley distribution, corresponding to the pdf given in equation (2.1) is

F(x;θ)=1-[ 1+ θx θ+1 ] e -θx ; x>0, θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=TdaiiaacqGF4oqCcaWFPaGaa8xpaiaa =fdacaWFTaGcdaWadaqaaKqzGeGaa8xmaiaa=TcakmaalaaabaqcLb sacqGF4oqCcaWF4baakeaajugibiab+H7aXjaa=TcacaWFXaaaaaGc caGLBbGaayzxaaqcLbsacaWFLbGcdaahaaWcbeqcbauaaKqzadGaa8 xlaiab+H7aXjaa=HhaaaqcLbsacaWFGaGaa8hiaiaa=TdacaWFGaGa a8hiaiaa=HhacaWF+aGaa8hmaiaa=XcacaWFGaGae4hUdeNaa8Npai aa=bdaaaa@5CBC@    (2.2)

This distribution is derived as a mixture of exponential (θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcaiiaacqGF4oqCcaWFPaaaaa@3C15@  and Gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajugibiaaikdacaGGSaaccaGae8hUdehakiaawIcacaGLPaaaaaa@3BCC@ distribution.

Hence the pdf takes the alternate form,

f(x;θ)=p f 1 (x)+(1-p) f 2 (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdaiiaacqGF4oqCcaWFPaGaa8xpaiaa =bhacaWFMbGcdaWgaaqcbauaaKqzadGaa8xmaaWcbeaajugibiaa=H cacaWF4bGaa8xkaiaa=TcacaWFOaGaa8xmaiaa=1cacaWFWbGaa8xk aiaa=zgakmaaBaaajeaqbaqcLbmacaWFYaaaleqaaKqzGeGaa8hkai aa=HhacaWFPaaaaa@50F5@

where β= θ 1+θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=j7aIHqaaiaa+1dakmaalaaabaqcLbsacqWF4oqCaOqaaKqzGeGa a4xmaiaa+TcacqWF4oqCaaaaaa@416B@  , f 1 (x)=θ e -θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgakmaaBaaajeaqbaqcLbmacaWFXaaaleqaaKqzGeGaa8hkaiaa =HhacaWFPaGaa8xpaGGaaiab+H7aXjaa=vgakmaaCaaaleqajeaqba qcLbmacaWFTaGae4hUdeNaa8hEaaaaaaa@478E@ , f 2 (x)= θ 2 Γ(2) e -θx x 2-1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgakmaaBaaajeaqbaqcLbmacaWFYaaaleqaaKqzGeGaa8hkaiaa =HhacaWFPaGaa8xpaOWaaSaaaeaaiiaajugibiab+H7aXPWaaWbaaS qabKqaafaajug4aiaa=jdaaaaakeaajugibiab+n5ahjaa=HcacaWF YaGaa8xkaaaacaWFLbGcdaahaaWcbeqcbauaaKqzadGaa8xlaiab+H 7aXjaa=HhaaaqcLbsacaWF4bGcdaahaaWcbeqcbauaaKqzadGaa8Nm aiaa=1cacaWFXaaaaaaa@5404@

and2 proved that in modeling and analysis of lifetime data, Lindley distribution provides a better model in many ways than the very usual exponential distribution. Along with various other properties they derived the central moments of Lindley distribution as,

μ 2 = θ 2 +4θ+2 θ 2 (θ+1) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=X7aTPWaaSbaaKqaafaaieaajugWaiaa+jdaaSqabaqcLbsacaGF GaGaa4xpaiaa+bcakmaalaaabaqcLbsacqWF4oqCkmaaCaaaleqaje aqbaqcLbmacaGFYaaaaKqzGeGaa43kaiaa+rdacqWF4oqCcaGFRaGa a4NmaaGcbaqcLbsacqWF4oqCkmaaCaaaleqajeaqbaqcLbmacaGFYa aaaKqzGeGaa4hkaiab=H7aXjaa+TcacaGFXaGaa4xkaOWaaWbaaSqa bKqaafaajugWaiaa+jdaaaaaaaaa@555F@

μ 3 = 2( θ 3 +6 θ 2 +6θ+2) θ 3 (θ+1) 3 and MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=X7aTPWaaSbaaKqaafaaieaajugWaiaa+ndaaSqabaqcLbsacaGF GaGaa4xpaiaa+bcakmaalaaabaqcLbsacaGFYaGaa4hkaiab=H7aXP WaaWbaaSqabKqaafaajugWaiaa+ndaaaqcLbsacaGFRaGaa4Nnaiab =H7aXPWaaWbaaSqabKqaafaajugWaiaa+jdaaaqcLbsacaGFRaGaa4 Nnaiab=H7aXjaa+TcacaGFYaGaa4xkaaGcbaqcLbsacqWF4oqCkmaa CaaaleqajeaqbaqcLbmacaGFZaaaaKqzGeGaa4hkaiab=H7aXjaa+T cacaGFXaGaa4xkaOWaaWbaaSqabKqaafaajugWaiaa+ndaaaaaaKqz GeGaa4hiaiaa+bcacaGFHbGaa4NBaiaa+rgaaaa@61EC@

μ 4 = 3(3 θ 4 +24 θ 3 +44 θ 2 +32θ+8) θ 4 (θ+1) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=X7aTPWaaSbaaKqaafaaieaajugWaiaa+rdaaSqabaqcLbsacaGF GaGaa4xpaiaa+bcakmaalaaabaqcLbsacaGFZaGaa4hkaiaa+ndacq WF4oqCkmaaCaaaleqajeaqbaqcLbmacaGF0aaaaKqzGeGaa43kaiaa +jdacaGF0aGae8hUdeNcdaahaaWcbeqcbauaaKqzadGaa43maaaaju gibiaa+TcacaGF0aGaa4hnaiab=H7aXPWaaWbaaSqabKqaafaajugW aiaa+jdaaaqcLbsacaGFRaGaa43maiaa+jdacqWF4oqCcaGFRaGaa4 hoaiaa+LcaaOqaaKqzGeGae8hUdeNcdaahaaWcbeqcbauaaKqzadGa a4hnaaaajugibiaa+HcacqWF4oqCcaGFRaGaa4xmaiaa+LcakmaaCa aaleqajeaqbaqcLbmacaGF0aaaaaaaaaa@663C@

Mazucheli3 discussed the applications of Lindley distribution on lifetime data regarding competing risks. Alternately, Shanker et al.4 made a comparison study of the goodness of fit of exponential and Lindley distributions on modeling of lifetime data. They provide different graphs for pdfs and cdfs for the same values of parameter for a visual comparison on the nature of the two distributions. They provide different associated functions as presented in Table 1.

The study associated fifteen different data sets and the fact revealed is that in some cases exponential distribution provides better fit than the Lindley distribution whereas in other cases Lindley distribution provides better fit than the exponential distribution. Thus4 does not made a final conclusion on the superiority of the two distributions. They arrive at a statement that the suitability depends on the nature of data. Exponential is simple, still Lindley is more flexible.

Quasi Lindley distribution

Shanker5 introduced Quasi Lindley distribution (QLD). QLD with parameters α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGajugibi aa=f7aaaa@3A3F@  and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7aaaa@3A44@  is defined by its pdf

f(x,α,θ)= θ(α+xθ) α+1 e -θx ,x>0,θ>0,α>-1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=XcaiiaacqGFXoqycaWFSaGae4hUdeNa a8xkaiaa=1dakmaalaaabaqcLbsacqGF4oqCcaWFOaGaa8xSdiaa=T cacaWF4bGae4hUdeNaa8xkaaGcbaqcLbsacaWFXoGaa83kaiaa=fda aaGaa8xzaOWaaWbaaSqabKqaafaajugWaiaa=1cacqGF4oqCcaWF4b aaaKqzGeGaaGjcVlaa=XcacaaMe8Uaa8hEaiaa=5dacaWFWaGaa8hl aiab+H7aXjaa=5dacaWFWaGaa8hlaiab+f7aHjaa=5dacaWFTaGaa8 xmaaaa@61E9@    (2.3)

It can be easily seen that at α=θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8hUdaaa@3C37@ , the QLD equation (2.3) reduces to the Lindley distribution given by equation (2.1) and at α=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8hmaaaa@3BAC@ , it reduces to the gamma distribution with parameters ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaG qaaKqzGeGaa8Nmaiaa=XcacaWF4oaakiaawIcacaGLPaaaaaa@3D37@ . The pdf equation (2.3) can be shown as a mixture of exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaG qaaKqzGeGaa8hUdaGccaGLOaGaayzkaaaaaa@3BD7@ and gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaG qaaKqzGeGaa8Nmaiaa=XcacaWF4oaakiaawIcacaGLPaaaaaa@3D37@ distributions as follows:

f(x,α,θ)=p f 1 (x)+(1-p) f 2 (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=XcaiiaacqGFXoqycaWFSaGae4hUdeNa a8xkaiaa=1dacaWFWbGaa8NzaOWaaSbaaKqaafaajugWaiaa=fdaaS qabaqcLbsacaWFOaGaa8hEaiaa=LcacaWFRaGaa8hkaiaa=fdacaWF TaGaa8hCaiaa=LcacaWFMbGcdaWgaaqcbauaaKqzadGaa8NmaaWcbe aajugibiaa=HcacaWF4bGaa8xkaaaa@532C@

where p= α α+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=bhacaWF9aGcdaWcaaqaaGGaaKqzGeGae4xSdegakeaajugibiab +f7aHjaa=TcacaWFXaaaaaaa@4093@ , f 1 (x)=θ e -θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgakmaaBaaajeaqbaqcLbmacaWFXaaaleqaaKqzGeGaa8hkaiaa =HhacaWFPaGaa8xpaiaa=H7acaWFLbGcdaahaaWcbeqcbauaaKqzad Gaa8xlaGGaaiab+H7aXjaa=Hhaaaaaaa@471A@  and f 2 (x)= θ 2 x e -θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgakmaaBaaajeaqbaqcLbmacaWFYaaaleqaaKqzGeGaa8hkaiaa =HhacaWFPaGaa8xpaGGaaiab+H7aXPWaaWbaaSqabKqaafaajugWai aa=jdaaaqcLbsacaWF4bGaa8xzaOWaaWbaaSqabKqaafaajugWaiaa =1cacqGF4oqCcaWF4baaaaaa@4B79@  The cdf of the QLD is obtained as

F(x,α,θ)=1- 1+α+θx α+1 e -θx ,x>0,θ>0,α>-1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=XcaiiaacqGFXoqycaWFSaGae4hUdeNa a8xkaiaa=1dacaWFXaGaa8xlaOWaaSaaaeaajugibiaa=fdacaWFRa Gae4xSdeMaa83kaiab+H7aXjaa=HhaaOqaaKqzGeGae4xSdeMaa83k aiaa=fdaaaGaa8xzaOWaaWbaaSqabKqaafaajugWaiaa=1cacqGF4o qCcaWF4baaaKqzGeGaaGjcVlaa=XcacaaMe8Uaa8hEaiaa=5dacaWF WaGaa8hlaiab+H7aXjaa=5dacaWFWaGaa8hlaiab+f7aHjaa=5daca WFTaGaa8xmaaaa@624C@

The QLD has been fitted to a number of data sets to which earlier the Lindley distribution has been fitted by others and it was found that to almost all these data‒sets, the QLD provides closer fits than those by the Lindley distribution.

Generalizations of Lindley distribution

Shanker et al.4 have comparative study on modeling of lifetime data using one parameter1 distribution and exponential distribution. Lindley distribution is not suitable for modeling data sets where there is large right tail or the tail approaches to zero at a faster rate. Such data sets are quite common in insurance problems and count data example in biology. This critical observation has motivated the authors to search for extensions of Lindley distribution. There exist many versions of generalized Lindley distribution and extensions. In this section, important among them are considered one by one.

Two parameter Lindley distribution

Shanker6 suggested a two parameter Lindley distribution (TPLD), of which the Lindley distribution is a particular case. There exist two forms type 1 and type 2. The pdf of type 1 is defined as

f(x;θ,α)= θ 2 αθ+1 (α+x) e -θx ;x>0 ,θ>0 ,αθ>-1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdaiiaacqGF4oqCcaWFSaGae4xSdeMa a8xkaiaa=1dakmaalaaabaqcLbsacqGF4oqCkmaaCaaaleqajeaqba qcLbmacaWFYaaaaaGcbaqcLbsacqGFXoqycqGF4oqCcaWFRaGaa8xm aaaacaWFOaGae4xSdeMaa83kaiaa=HhacaWFPaGaa8xzaOWaaWbaaS qabKqaafaajugWaiaa=1cacqGF4oqCcaWF4baaaKqzGeGaa8hiaiaa =bcacaWF7aGaa8hEaiaa=5dacaWFWaGaa8hiaiaa=XcacqGF4oqCca WF+aGaa8hmaiaa=bcacaWFSaGae4xSdeMae4hUdeNaa8Npaiaa=1ca caWFXaaaaa@6647@

When α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8xmaaaa@3BAD@ , it gives the Lindley distribution and α=θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8hUdaaa@3C37@ , it gives the gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaG qaaKqzGeGaa8Nmaiaa=XcacaWF4oaakiaawIcacaGLPaaaaaa@3D37@ distribution. The pdf is a mixture of exponential (θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWF4oGaa8xkaaaa@3B97@  and gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqadaqaaG qaaKqzGeGaa8Nmaiaa=XcacaWF4oaakiaawIcacaGLPaaaaaa@3D37@ distribution where the mixing constant is αθ αθ+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG GaaKqzGeGae8xSdeMae8hUdehakeaajugibiab=f7aHjab=H7aXHqa aiaa+TcacaGFXaaaaaaa@41AD@ .

The cdf of type 1 TPLD is,

F(x:α,θ)=1- 1+αθ+θx αθ+1 e -θx ; x>0, θ>0, αθ>-1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=PdaiiaacqGFXoqycaWFSaGae4hUdeNa a8xkaiaa=1dacaWFXaGaa8xlaOWaaSaaaeaajugibiaa=fdacaWFRa Gae4xSdeMae4hUdeNaa83kaiab+H7aXjaa=HhaaOqaaKqzGeGae4xS deMae4hUdeNaa83kaiaa=fdaaaGaa8xzaOWaaWbaaSqabKqaafaaju gWaiaa=1cacqGF4oqCcaWF4baaaKqzGeGaa8hiaiaa=TdacaWFGaGa a8hEaiaa=5dacaWFWaGaa8hlaiaa=bcacqGF4oqCcaWF+aGaa8hmai aa=XcacaWFGaGae4xSdeMae4hUdeNaa8Npaiaa=1cacaWFXaaaaa@66DF@

Then the corresponding failure rate function h(x) is

h(x)= θ 2 (1+αx) θ+α+θαx ; x>0, θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaGGaaKqzGeGa e4hUdeNcdaahaaWcbeqcbauaaKqzadGaa8Nmaaaajugibiaa=Hcaca WFXaGaa83kaiab+f7aHjaa=HhacaWFPaaakeaajugibiab+H7aXjaa =TcacqGFXoqycaWFRaGae4hUdeNae4xSdeMaa8hEaaaacaWF7aGaa8 hiaiaa=HhacaWF+aGaa8hmaiaa=XcacaWFGaGae4hUdeNaa8Npaiaa =bdaaaa@5942@

The pdf of type 2 is defined as

f(x;θ,β)= θ 2 θ+β (1+βx) e -θx ;x>0 ,θ,β,>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdaiiaacqGF4oqCcaWFSaGae4NSdiMa a8xkaiaa=1dakmaalaaabaqcLbsacqGF4oqCkmaaCaaaleqajeaqba qcLbmacaWFYaaaaaGcbaqcLbsacqGF4oqCcaWFRaGae4NSdigaaiaa =HcacaWFXaGaa83kaiab+j7aIjaa=HhacaWFPaGaa8xzaOWaaWbaaS qabKqaafaajugWaiaa=1cacqGF4oqCcaWF4baaaKqzGeGaa8hiaiaa =bcacaWF7aGaa8hEaiaa=5dacaWFWaGaa8hiaiaa=XcacqGF4oqCca WFSaGae4NSdiMaa8hlaiaa=5dacaWFWaaaaa@628C@

The cdf of type 2 TPLD is,

F(x:β,θ)=1- θ+β+βθx θ+β e -θx ; x>0, θ>0, β>-θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=PdaiiaacqGFYoGycaWFSaGae4hUdeNa a8xkaiaa=1dacaWFXaGaa8xlaOWaaSaaaeaajugibiab+H7aXjaa=T cacqGFYoGycaWFRaGae4NSdiMae4hUdeNaa8hEaaGcbaqcLbsacqGF 4oqCcaWFRaGae4NSdigaaiaa=vgakmaaCaaaleqajeaqbaqcLbmaca WFTaGaa8hUdiaa=HhaaaqcLbsacaWFGaGaa83oaiaa=bcacaWF4bGa a8Npaiaa=bdacaWFSaGaa8hiaiab+H7aXjaa=5dacaWFWaGaa8hlai aa=bcacqGFYoGycaWF+aGaa8xlaiab+H7aXbaa@65F8@

Then the corresponding failure rate function h(x) is

h(x)= θ 2 (1+βx) θ+β+θβx ; x>0, θ>0, β>-θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaGGaaKqzGeGa e4hUdeNcdaahaaWcbeqcbauaaKqzadGaa8Nmaaaajugibiaa=Hcaca WFXaGaa83kaiab+j7aIjaa=HhacaWFPaaakeaajugibiab+H7aXjaa =TcacqGFYoGycaWFRaGae4hUdeNae4NSdiMaa8hEaaaacaWFGaGaa8 3oaiaa=bcacaWF4bGaa8Npaiaa=bdacaWFSaGaa8hiaiab+H7aXjaa =5dacaWFWaGaa8hlaiaa=bcacqGFYoGycaWF+aGaa8xlaiab+H7aXb aa@5FEF@

The mean of TPLD is always greater than the mode, the distribution is positively skewed. The TPLD provides better fits than those by the one parameter Lindley distribution.7 Obtained a two parameter weighted Lindley distribution and studied its applications to survival data.8 Studied two parameter Lindley distribution for modeling waiting and survival data.9 Proposed an extended Lindley distribution which offers a more flexible model for lifetime data.

Akash distribution

The one parameter lifetime distribution with pdf

f(x,θ)= θ 3 θ 2 +2 (1+ x 2 ) e -θx ; x>0 ,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=XcaiiaacqGF4oqCcaWFPaGaa8xpaOWa aSaaaeaajugibiab+H7aXPWaaWbaaSqabKqaGfaajugWaiaa=ndaaa aakeaajugibiab+H7aXPWaaWbaaSqabKqaafaajugWaiaa=jdaaaqc LbsacaWFRaGaa8NmaaaacaWFOaGaa8xmaiaa=TcacaWF4bGcdaahaa WcbeqcbauaaKqzadGaa8Nmaaaajugibiaa=LcacaWFLbGcdaahaaWc beqcbauaaKqzadGaa8xlaiab+H7aXjaa=HhaaaqcLbsacaWFGaGaa8 3oaiaa=bcacaWF4bGaa8Npaiaa=bdacaWFGaGaa8hlaiab+H7aXjaa =5dacaWFWaaaaa@60C0@

Suggested by Shanker10 is termed as Akash distribution. The pdf is a mixture of exponential (θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWF4oGaa8xkaaaa@3B97@ and gamma (3,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFZaGaa8hlaiaa=H7acaWFPaaaaa@3CF8@  distribution where the mixing constant θ 2 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG GaaKqzGeGae8hUdeNcdaahaaWcbeqcbauaaGqaaKqzadGaa4Nmaaaa aOqaaKqzGeGae8hUdeNcdaahaaWcbeqcbauaaKqzadGaa4Nmaaaaju gibiaa+TcacaGFYaaaaaaa@43CB@ . The cdf of Akash distribution is

F(x)=1-[ 1+ θx(θx+2 θ 2 +2 ] e -θx ; x>0, θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=LcacaWF9aGaa8xmaiaa=1cakmaadmaa baqcLbsacaWFXaGaa83kaOWaaSaaaeaaiiaajugibiab+H7aXjaa=H hacaWFOaGae4hUdeNaa8hEaiaa=TcacaWFYaaakeaajugibiab+H7a XPWaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFRaGaa8Nmaa aaaOGaay5waiaaw2faaKqzGeGaa8xzaOWaaWbaaSqabKqaafaajugW aiaa=1cacqGF4oqCcaWF4baaaKqzGeGaa8hiaiaa=TdacaWFGaGaa8 hEaiaa=5dacaWFWaGaa8hlaiaa=bcacqGF4oqCcaWF+aGaa8hmaaaa @60B1@

and the corresponding hazard rate function h(x) is

h(x)= θ 2 (1+x) (θ+1)+θx ; x>0, θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaGGaaKqzGeGa e4hUdeNcdaahaaWcbeqcbauaaKqzadGaa8Nmaaaajugibiaa=Hcaca WFXaGaa83kaiaa=HhacaWFPaaakeaajugibiaa=HcacqGF4oqCcaWF RaGaa8xmaiaa=LcacaWFRaGae4hUdeNaa8hEaaaacaWF7aGaa8hiai aa=HhacaWF+aGaa8hmaiaa=XcacaWFGaGae4hUdeNaa8Npaiaa=bda aaa@567C@

Akash distribution is over‒dispersed (when μ> σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=X7aTHqaaiaa+5dacqWFdpWCkmaaCaaaleqajeaqbaqcLbmacaGF Yaaaaaaa@3FA0@ ), equi‒dispersed (when μ> σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=X7aTHqaaiaa+5dacqWFdpWCkmaaCaaaleqajeaqbaqcLbmacaGF Yaaaaaaa@3FA0@ ) and under‒dispersed (when μ< σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=X7aTHqaaiaa+XdacqWFdpWCkmaaCaaaleqajeaqbaqcLbmacaGF Yaaaaaaa@3F9E@ ). The hazard rate function of Akash distribution is an increasing function of x and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7aaaa@3A44@ . Also it is better than Lindley and exponential distribution for modeling life time data from medical science and engineering. Shanker11 has introduced a quasi Akash distribution for modeling lifetime data and discussed its statistical properties and applications. Also11 have comparative study on lifetime data using one parameter Akash, Lindley and exponential distribution and showed that Akash distribution gives better fit in some of the data set than the other two distributions.

Two parameter akash distribution

Shanker12 proposed a two parameter Akash distribution (TPAD) with parameters θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7aaaa@3A44@ and α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7aaaa@3A3D@  is defined by its pdf

f(x;θ,α)= θ 3 α θ 2 +2 (α+ x 2 ) e -θx ; x>0 ,θ>0, α>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdaiiaacqGF4oqCcaWFSaGae4xSdeMa a8xkaiaa=1dakmaalaaabaqcLbsacqGF4oqCkmaaCaaaleqajeaqba qcLbmacaWFZaaaaaGcbaqcLbsacqGFXoqycqGF4oqCkmaaCaaaleqa jeaqbaqcLbmacaWFYaaaaKqzGeGaa83kaiaa=jdaaaGaa8hkaiab+f 7aHjaa=TcacaWF4bGcdaahaaWcbeqcbauaaKqzadGaa8Nmaaaajugi biaa=LcacaWFLbGcdaahaaWcbeqcbauaaKqzadGaa8xlaiab+H7aXj aa=HhaaaqcLbsacaWFGaGaa83oaiaa=bcacaWF4bGaa8Npaiaa=bda caWFGaGaa8hlaiab+H7aXjaa=5dacaWFWaGaa8hlaiaa=bcacqGFXo qycaWF+aGaa8hmaaaa@69CC@

At α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8xmaaaa@3BAD@ , it gives the Akash distribution. The pdf of TPAD is a convex combination of exponential (θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWF4oGaa8xkaaaa@3B97@  and gamma (3,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFZaGaa8hlaiaa=H7acaWFPaaaaa@3CF8@  with mixing constant α θ 2 α θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG GaaKqzGeGae8xSdeMae8hUdeNcdaahaaWcbeqcbauaaGqaaKqzadGa a4NmaaaaaOqaaKqzGeGae8xSdeMae8hUdeNcdaahaaWcbeqcbauaaK qzadGaa4Nmaaaajugibiaa+TcacaGFYaaaaaaa@46FF@ . Its cdf

F(x)=1-[ 1+ θx(θx+2 α θ 2 +2 ] e -θx ; x>0, θ>0, α>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=LcacaWF9aGaa8xmaiaa=1cakmaadmaa baqcLbsacaWFXaGaa83kaOWaaSaaaeaaiiaajugibiab+H7aXjaa=H hacaWFOaGae4hUdeNaa8hEaiaa=TcacaWFYaaakeaajugibiab+f7a Hjab+H7aXPWaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFRa Gaa8NmaaaaaOGaay5waiaaw2faaKqzGeGaa8xzaOWaaWbaaSqabKqa afaajugWaiaa=1cacqGF4oqCcaWF4baaaKqzGeGaa8hiaiaa=Tdaca WFGaGaa8hEaiaa=5dacaWFWaGaa8hlaiaa=bcacqGF4oqCcaWF+aGa a8hmaiaa=XcacaWFGaGae4xSdeMaa8Npaiaa=bdaaaa@66A1@

The corresponding failure rate function or hazard rate function h(x) is

h(x)= θ 3 (α+ x 2 ) θx(θx+2)+(α θ 2 +2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaGGaaKqzGeGa e4hUdeNcdaahaaWcbeqcbauaaKqzadGaa83maaaajugibiaa=Hcacq GFXoqycaWFRaGaa8hEaOWaaWbaaSqabKqaafaajugWaiaa=jdaaaqc LbsacaWFPaaakeaajugibiab+H7aXjaa=HhacaWFOaGae4hUdeNaa8 hEaiaa=TcacaWFYaGaa8xkaiaa=TcacaWFOaGae4xSdeMae4hUdeNc daahaaWcbeqcbauaaKqzadGaa8Nmaaaajugibiaa=TcacaWFYaGaa8 xkaaaaaaa@5C07@

TPAD gives better fit over exponential, Akash and Lognormal distributions. Shanker12 studied weighted Akash distribution for modelling lifetime data and observed that it gives better fit than several one parameter and two parameter lifetime distribution.

Generalized Lindley distribution

Zakerzadeh13 suggested a generalized Lindley distribution (GLD), of which the Lindley distribution is a particular case. The pdf is defined as

f(x;α,θ,γ)= θ 2 (θx) α-1 (α+γx) e -θx (γ+θ)Γ(α+1) ; α, θ, γ, x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdaiiaacqGFXoqycaWFSaGae4hUdeNa a8hlaiab+n7aNjaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hUdOWaaW baaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFOaGaa8hUdiaa=Hha caWFPaGcdaahaaWcbeqcbauaaKqzadGae4xSdeMaa8xlaiaa=fdaaa qcLbsacaWFOaGae4xSdeMaa83kaiab+n7aNjaa=HhacaWFPaGaa8xz aOWaaWbaaSqabKqaafaajugWaiaa=1cacaWF4oGaa8hEaaaaaOqaaK qzGeGaa8hkaiab+n7aNjaa=TcacaWF4oGaa8xkaiab+n5ahjaa=Hca cqGFXoqycaWFRaGaa8xmaiaa=LcaaaGaa8hiaiaa=TdacaWFGaGae4 xSdeMaa8hlaiaa=bcacaWF4oGaa8hlaiaa=bcacqGFZoWzcaWFSaGa a8hiaiaa=HhacaWF+aGaa8hmaiaa=bcaaaa@75CA@

That is this is a three parameter distribution. At α=γ=1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=f7aHjab=1da9iab=n7aNjab=1da9Gqaaiaa+fdacaGFSaaaaa@3FA9@  it gives the Lindley distribution and γ=0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=n7acaWF9aGaa8hmaiaa=Xcaaaa@3C5B@  it gives the gamma (α,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFXoGaa8hlaiaa=H7acaWFPaaaaa@3D79@  distribution .The pdf is a mixture of gamma (α,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFXoGaa8hlaiaa=H7acaWFPaaaaa@3D79@ and gamma (α+1,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcaiiaacqGFXoqycaWFRaGaa8xmaiaa=XcacqGF4oqCcaWFPaaa aa@3FB9@  distribution where the mixing constant is θ γ+θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG Gaaiab=H7aXbqaaiab=n7aNHqaaiaa+TcacqWF4oqCaaaaaa@3E42@ .

The cdf of Generalized Lindley distribution can be given only in terms of the incomplete gamma function when α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7aaaa@3A3D@  is not an integer, the hazard function α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7aaaa@3A3D@ could not be expressed in closed form.

A new generalized Lindley distribution

Ibrahim et al.14 proposed a new generalized Lindley distribution (NGLD) is obtained from a mixture of the gamma (α,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFXoGaa8hlaiaa=H7acaWFPaaaaa@3D79@  and gamma (β,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFYoGaa8hlaiaa=H7acaWFPaaaaa@3D7A@  where the mixing constant is θ 1+θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG GaaKqzGeGae8hUdehakeaaieaajugibiaa+fdacaGFRaGae8hUdeha aaaa@3E79@ . The pdf of NGLD is

f(x;θ,α,β)= 1 1+θ [ θ α+1 x α-1 Γ(α) + θ β x β-1 Γ(β) ] e -θx ; α, θ>0 x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdaiiaacqGF4oqCcaWFSaGae4xSdeMa a8hlaiab+j7aIjaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xmaaGcba qcLbsacaWFXaGaa83kaiab+H7aXbaakmaadmaabaWaaSaaaeaajugi biab+H7aXPWaaWbaaSqabKqaafaajugWaiab+f7aHjaa=TcacaWFXa aaaKqzGeGaa8hEaOWaaWbaaSqabKqaafaajugWaiab+f7aHjaa=1ca caWFXaaaaaGcbaqcLbsacaWFtoGaa8hkaiab+f7aHjaa=LcaaaGaa8 3kaOWaaSaaaeaajugibiab+H7aXPWaaWbaaSqabKqaafaajugWaiab +j7aIbaajugibiaa=HhakmaaCaaaleqajeaqbaqcLbmacqGFYoGyca WFTaGaa8xmaaaaaOqaaKqzGeGaa83Kdiaa=HcacqGFYoGycaWFPaaa aaGccaGLBbGaayzxaaqcLbsacaWFLbGcdaahaaWcbeqcbauaaKqzad Gaa8xlaiab+H7aXjaa=HhaaaqcLbsacaWFGaGaa83oaiaa=bcacqGF XoqycaWFSaGaa8hiaiaa=bcacqGF4oqCcaWF+aGaa8hmaiaa=bcaca WF4bGaa8Npaiaa=bdaaaa@809F@

The corresponding cdf is given by

F(x;θ,α,β)= 1 1+θ [ θγ(α,θx) Γ(α) + γ(β,θx) Γ(β) ] ; α, θ>0 x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=TdaiiaacqGF4oqCcaWFSaGae4xSdeMa a8hlaiab+j7aIjaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xmaaGcba qcLbsacaWFXaGaa83kaiab+H7aXbaakmaadmaabaWaaSaaaeaajugi biab+H7aXjaa=n7acaWFOaGaa8xSdiaa=XcacqGF4oqCcaWF4bGaa8 xkaaGcbaqcLbsacaWFtoGaa8hkaiab+f7aHjaa=LcaaaGaa83kaOWa aSaaaeaajugibiaa=n7acaWFOaGae4NSdiMaa8hlaiab+H7aXjaa=H hacaWFPaaakeaajugibiaa=n5acaWFOaGae4NSdiMaa8xkaaaaaOGa ay5waiaaw2faaKqzGeGaa8hiaiaa=TdacaWFGaGae4xSdeMaa8hlai aa=bcacqGF4oqCcaWF+aGaa8hmaiaa=bcacaWF4bGaa8Npaiaa=bda aaa@71F1@

The hazard rate function h(x) is

h(x)= 1 1+θ [ θ α+1 x α-1 Γ(α) + θ β x β-1 Γ(β) ] e -θx 1- 1 1+θ [ θγ(α,θx) Γ(α) + γ(β,θx) Γ(β) ] ;α, θ>0 x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaamaalaaabaqc LbsacaWFXaaakeaajugibiaa=fdacaWFRaGaa8hUdaaakmaadmaaba WaaSaaaeaajugibiaa=H7akmaaCaaaleqajeaqbaqcLbmacaWFXoGa a83kaiaa=fdaaaqcLbsacaWF4bGcdaahaaWcbeqcbauaaKqzadGaa8 xSdiaa=1cacaWFXaaaaaGcbaqcLbsacaWFtoGaa8hkaiaa=f7acaWF Paaaaiaa=TcakmaalaaabaqcLbsacaWF4oGcdaahaaWcbeqcbauaaK qzadGaa8NSdaaajugibiaa=HhakmaaCaaaleqajeaqbaqcLbmacaWF YoGaa8xlaiaa=fdaaaaakeaajugibiaa=n5acaWFOaGaa8NSdiaa=L caaaaakiaawUfacaGLDbaajugibiaa=vgakmaaCaaaleqajeaqbaqc LbmacaWFTaGaa8hUdiaa=Hhaaaaakeaajugibiaa=fdacaWFTaGcda WcaaqaaKqzGeGaa8xmaaGcbaqcLbsacaWFXaGaa83kaiaa=H7aaaGc daWadaqaamaalaaabaqcLbsacaWF4oGaa83Sdiaa=HcacaWFXoGaa8 hlaiaa=H7acaWF4bGaa8xkaaGcbaqcLbsacaWFtoGaa8hkaiaa=f7a caWFPaaaaiaa=TcakmaalaaabaqcLbsacaWFZoGaa8hkaiaa=j7aca WFSaGaa8hUdiaa=HhacaWFPaaakeaajugibiaa=n5acaWFOaGaa8NS diaa=LcaaaaakiaawUfacaGLDbaaaaqcLbsacaWFGaGaa83oaiaa=f 7acaWFSaGaa8hiaiaa=H7acaWF+aGaa8hmaiaa=bcacaWF4bGaa8Np aiaa=bdaaaa@94D9@

The hazard rate function of GLD, is increasing for α1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7aiiaacqGFLjYScaWFXaaaaa@3CB7@ , bathtub shaped for α<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF8aGaa8xmaaaa@3BAC@ and γ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=n7acaWF+aGaa8hmaaaa@3BAF@ , decreasing for α1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7aiiaacqGFKjYOcaWFXaaaaa@3CA6@  and γ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=n7acaWF9aGaa8hmaaaa@3BAE@ . At α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8xmaaaa@3BAD@  and β=2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=j7acaWF9aGaa8Nmaaaa@3BAF@ , the NGLD becomes Lindley distribution, for α=β=λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8NSdiaa=1dacaWF7oaaaa@3E2E@ , NGLD becomes gamma distribution with parameter (θ,λ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWF4oGaa8hlaiaa=T7acaWFPaaaaa@3D83@  and at α=β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8NSdiaa=1dacaWFXaaaaa@3DA1@  it becomes exponential distribution with parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7aaaa@3A44@ . Gupta15 studied generalized exponential distribution. The Lindley distribution has been generalized by different researchers including.16‒21

Power Lindley distribution

Ghitany et al.22 introduced Power Lindley distribution (PLD) with pdf

f(x;θ,α)= α θ 2 θ+1 (1+ x α ) x α-1 e -θ x α ;θ,α,x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdacaWF4oGaa8hlaiaa=f7acaWFPaGa a8xpaOWaaSaaaeaajugibiaa=f7acaWF4oGcdaahaaWcbeqcbauaaK qzadGaa8NmaaaaaOqaaKqzGeGaa8hUdiaa=TcacaWFXaaaaiaa=Hca caWFXaGaa83kaiaa=HhakmaaCaaaleqajeaqbaqcLbmacaWFXoaaaK qzGeGaa8xkaiaa=HhakmaaCaaaleqajeaqbaqcLbmacaWFXoGaa8xl aiaa=fdaaaqcLbsacaWFLbGcdaahaaWcbeqcbauaaKqzadGaa8xlai aa=H7acaWF4bqcfa4aaWbaaKqaafqabaqcLbmacaWFXoaaaaaajugi biaa=bcacaWFGaGaa83oaiaa=H7acaWFSaGaa8xSdiaa=XcacaWF4b Gaa8Npaiaa=bdacaWFGaaaaa@68A2@

It is a new extension of Lindley distribution by considering the power transformation of the random variable X= Y 1 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HfacaWF9aGaa8xwaOWaaWbaaSqabeaakmaaCaaaleqajeaqbaqc fa4aaSaaaKqaafaajugWaiaa=fdaaKqaafaajugWaiaa=f7aaaaaaa aaaaa@41A6@ , where Y follows Lindley distribution. At α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8xmaaaa@3BAD@  it reduces to Lindley distribution. It is a mixture of Weibull distribution (α,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFXoGaa8hlaiaa=H7acaWFPaaaaa@3D79@  and generalized gamma distribution (2α,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFYaGaa8xSdiaa=XcacaWF4oGaa8xkaaaa@3E2C@  with mixing proportion θ 1+θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8hUdaGcbaqcLbsacaWFXaGaa83kaiaa=H7aaaaaaa@3D87@ .

The corresponding cdf is given by

F(x;θ,α)=1- θ+1+θ x α θ+1 e -θ x α ;θ,α,x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=TdacaWF4oGaa8hlaiaa=f7acaWFPaGa a8xpaiaa=fdacaWFTaGcdaWcaaqaaKqzGeGaa8hUdiaa=TcacaWFXa Gaa83kaiaa=H7acaWF4bGcdaahaaWcbeqcbauaaKqzadGaa8xSdaaa aOqaaKqzGeGaa8hUdiaa=TcacaWFXaaaaiaa=vgakmaaCaaaleqaje aqbaqcLbmacaWFTaGaa8hUdiaa=HhajuaGdaahaaqcbauabeaajugW aiaa=f7aaaaaaKqzGeGaa8hiaiaa=bcacaWF7aGaa8hUdiaa=Xcaca WFXoGaa8hlaiaa=HhacaWF+aGaa8hmaiaa=bcaaaa@6085@

The associated hazard rate function is

h(x)= α θ 2 x α-1 (1+ x α θ+1+θ x α ) ;θ,α,x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xS diaa=H7akmaaCaaaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa8hEaO WaaWbaaSqabKqaafaajugWaiaa=f7acaWFTaGaa8xmaaaajugibiaa =HcacaWFXaGaa83kaiaa=HhakmaaCaaaleqajeaqbaqcLbmacaWFXo aaaaGcbaqcLbsacaWF4oGaa83kaiaa=fdacaWFRaGaa8hUdiaa=Hha kmaaCaaaleqajeaqbaqcLbmacaWFXoaaaKqzGeGaa8xkaaaacaWF7a Gaa8hUdiaa=XcacaWFXoGaa8hlaiaa=HhacaWF+aGaa8hmaaaa@5FD7@

Extended Power Lindley Distribution

Alkarni23 suggested an Extended Power Lindley Distribution with parameters θ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7acaWFSaGaa8xSdaaa@3C26@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=j7aaaa@3A3E@  is defined by its pdf

f(x;θ,β,α)= α θ 2 θ+β (1+β x α ) x α-1 e -θ x α ;θ,β,α,x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdacaWF4oGaa8hlaiaa=j7acaWFSaGa a8xSdiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xSdiaa=H7akmaaCa aaleqajeaqbaqcLbmacaWFYaaaaaGcbaqcLbsacaWF4oGaa83kaiaa =j7aaaGaa8hkaiaa=fdacaWFRaGaa8NSdiaa=HhakmaaCaaaleqaje aqbaqcLbmacaWFXoaaaKqzGeGaa8xkaiaa=HhakmaaCaaaleqajeaq baqcLbmacaWFXoGaa8xlaiaa=fdaaaqcLbsacaWFLbGcdaahaaWcbe qcbauaaKqzadGaa8xlaiaa=H7acaWF4bqcfa4aaWbaaKqaafqabaqc LbmacaWFXoaaaaaajugibiaa=bcacaWFGaGaa83oaiaa=H7acaWFSa Gaa8NSdiaa=XcacaWFXoGaa8hlaiaa=HhacaWF+aGaa8hmaiaa=bca aaa@6E22@

It is a extension of type 2 TPLD by considering the power transformation of the random variable X= Y 1 α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HfacaWF9aGaa8xwaOWaaWbaaSqabeaakmaaCaaaleqajeaqbaqc fa4aaSaaaKqaafaajugWaiaa=fdaaKqaafaajugWaiaa=f7aaaaaaa aaaaa@41A6@ , where Y follows type 2 TPLD. It is a mixture of Weibull distribution (α,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFXoGaa8hlaiaa=H7acaWFPaaaaa@3D79@  and generalized gamma distribution (2α,θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HcacaWFYaGaa8xSdiaa=XcacaWF4oGaa8xkaaaa@3E2C@  with mixing proportion θ θ+α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8hUdaGcbaqcLbsacaWF4oGaa83kaiaa=f7aaaaaaa@3E0A@ . At α=β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8NSdiaa=1dacaWFXaaaaa@3DA1@ . It gives the Lindley distribution, at α=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=f7acaWF9aGaa8xmaaaa@3BAD@  it gives TPLD, at β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=j7acaWF9aGaa8xmaaaa@3BAE@  it gives power distribution and at γ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=n7acaWF9aGaa8hmaaaa@3BAE@ , it gives Weibull distribution.

The corresponding cdf is given by

F(x;θ,β,α)=1-( 1+ θβ θ+β x α ) e -θ x α ; θ, β, α, x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=TdacaWF4oGaa8hlaiaa=j7acaWFSaGa a8xSdiaa=LcacaWF9aGaa8xmaiaa=1cakmaabmaabaqcLbsacaWFXa Gaa83kaOWaaSaaaeaajugibiaa=H7acaWFYoaakeaajugibiaa=H7a caWFRaGaa8NSdaaacaWF4bGcdaahaaWcbeqcbauaaKqzadGaa8xSda aaaOGaayjkaiaawMcaaKqzGeGaa8xzaOWaaWbaaSqabKqaafaajugW aiaa=1cacaWF4oGaa8hEaKqbaoaaCaaajeaqbeqaaKqzadGaa8xSda aaaaqcLbsacaWFGaGaa83oaiaa=bcacaWF4oGaa8hlaiaa=bcacaWF YoGaa8hlaiaa=bcacaWFXoGaa8hlaiaa=bcacaWF4bGaa8Npaiaa=b dacaWFGaaaaa@68BB@

The associated hazard rate function is

h(x)= α θ 2 x α-1 (1+β x α ) θ+β+βθ x α ;θ, β, α, x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xS diaa=H7akmaaCaaaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa8hEaO WaaWbaaSqabKqaafaajugWaiaa=f7acaWFTaGaa8xmaaaajugibiaa =HcacaWFXaGaa83kaiaa=j7acaWF4bGcdaahaaWcbeqcbauaaKqzad Gaa8xSdaaajugibiaa=LcaaOqaaKqzGeGaa8hUdiaa=TcacaWFYoGa a83kaiaa=j7acaWF4oGaa8hEaOWaaWbaaSqabKqaafaajugWaiaa=f 7aaaaaaKqzGeGaa8hiaiaa=bcacaWF7aGaa8hUdiaa=XcacaWFGaGa a8NSdiaa=XcacaWFGaGaa8xSdiaa=XcacaWFGaGaa8hEaiaa=5daca WFWaGaa8hiaaaa@68FF@

h(x) is increasing for all θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7acaWF+aGaa8hmaaaa@3BB4@ .

Exponentiated power Lindley distribution

Ashour24 suggested an Exponentiated Power Lindley Distribution with parameters θ,α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWF4o Gaa8hlaiaa=f7aaaa@3B97@  and β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=j7aaaa@3A3E@  is defined by its pdf

f(x;θ,β,α)= α θ 2 β x β-1 θ+1 (1+ x β ) e -θ x β [ 1-( 1+ θ x β θ+1 ) e -θ x β ] α-1 ;θ,β,α,x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdacaWF4oGaa8hlaiaa=j7acaWFSaGa a8xSdiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xSdiaa=H7akmaaCa aaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa8NSdiaa=HhakmaaCaaa leqajeaqbaqcLbmacaWFYoGaa8xlaiaa=fdaaaaakeaajugibiaa=H 7acaWFRaGaa8xmaaaacaWFOaGaa8xmaiaa=TcacaWF4bGcdaahaaWc beqcbauaaKqzadGaa8NSdaaajugibiaa=LcacaWFLbGcdaahaaWcbe qcbauaaKqzadGaa8xlaiaa=H7acaWF4bqcfa4aaWbaaKqaafqabaqc LbmacaWFYoaaaaaakmaadmaabaqcLbsacaWFXaGaa8xlaOWaaeWaae aajugibiaa=fdacaWFRaGcdaWcaaqaaKqzGeGaa8hUdiaa=Hhakmaa CaaaleqajeaqbaqcLbmacaWFYoaaaaGcbaqcLbsacaWF4oGaa83kai aa=fdaaaaakiaawIcacaGLPaaajugibiaa=vgakmaaCaaaleqajeaq baqcLbmacaWFTaGaa8hUdiaa=HhajuaGdaahaaqcbauabeaajugWai aa=j7aaaaaaaGccaGLBbGaayzxaaWaaWbaaSqabKqaafaajugWaiaa =f7acaWFTaGaa8xmaaaajugibiaa=bcacaWFGaGaa83oaiaa=H7aca WFSaGaa8NSdiaa=XcacaWFXoGaa8hlaiaa=HhacaWF+aGaa8hmaiaa =bcaaaa@8BB5@

For β=1,α=1,and α=β=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=j7acaWF9aGaa8xmaiaaysW7caWFSaGaa8xSdiaa=1dacaWFXaGa a8hlaiaa=fgacaWFUbGaa8hzaiaa=bcacaWFXoGaa8xpaiaa=j7aca WF9aGaa8xmaaaa@492A@ , It gives the generalized Lindley distribution, Power Lindley and Lindley distribution. The corresponding cdf is given by

F(x;θ,β,α)= [ 1-( 1+ θ x β θ+1 ) e -θ x β ] α ;θ, β, α, x>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=TdacaWF4oGaa8hlaiaa=j7acaWFSaGa a8xSdiaa=LcacaWF9aGcdaWadaqaaKqzGeGaa8xmaiaa=1cakmaabm aabaqcLbsacaWFXaGaa83kaOWaaSaaaeaajugibiaa=H7acaWF4bGc daahaaWcbeqcbauaaKqzadGaa8NSdaaaaOqaaKqzGeGaa8hUdiaa=T cacaWFXaaaaaGccaGLOaGaayzkaaqcLbsacaWFLbGcdaahaaWcbeqc bauaaKqzadGaa8xlaiaa=H7acaWF4bqcfa4aaWbaaKqaafqabaqcLb macaWFYoaaaaaaaOGaay5waiaaw2faamaaCaaaleqajeaqbaqcLbma caWFXoaaaKqzGeGaa8hiaiaa=bcacaWF7aGaa8hUdiaa=XcacaWFGa Gaa8NSdiaa=XcacaWFGaGaa8xSdiaa=XcacaWFGaGaa8hEaiaa=5da caWFWaGaa8hiaaaa@6C72@

The associated hazard rate function is

h(x)= α θ 2 β x β-1 θ+1 (1+ x β ) e -θ x β [ 1-( 1+ θ x β θ+1 ] e -θ x β ] α-1 S (x) -1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xS diaa=H7akmaaCaaaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa8NSdi aa=HhakmaaCaaaleqajeaqbaqcLbmacaWFYoGaa8xlaiaa=fdaaaaa keaajugibiaa=H7acaWFRaGaa8xmaaaacaWFOaGaa8xmaiaa=Tcaca WF4bGcdaahaaWcbeqcbauaaKqzadGaa8NSdaaajugibiaa=LcacaWF LbGcdaahaaWcbeqcbauaaKqzadGaa8xlaiaa=H7acaWF4bqcfa4aaW baaKqaafqabaqcLbmacaWFYoaaaaaakmaadmaabaqcLbsacaWFXaGa a8xlaOWaaKamaeaajugibiaa=fdacaWFRaGcdaWcaaqaaKqzGeGaa8 hUdiaa=HhakmaaCaaaleqajeaqbaqcLbmacaWFYoaaaaGcbaqcLbsa caWF4oGaa83kaiaa=fdaaaaakiaawIcacaGLDbaajugibiaa=vgakm aaCaaaleqajeaqbaqcLbmacaWFTaGaa8hUdiaa=HhajuaGdaahaaqc bauabeaajugWaiaa=j7aaaaaaaGccaGLBbGaayzxaaWaaWbaaSqabK qaafaajugWaiaa=f7acaWFTaGaa8xmaaaajugibiaa=nfacaWFOaGa a8hEaiaa=LcakmaaCaaaleqajeaqbaqcLbmacaWFTaGaa8xmaaaaaa a@81DC@

Where

S(x)=1- [ 1-( 1+ θ x β θ+1 ) e -θ x β ] α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=nfacaWFOaGaa8hEaiaa=LcacaWF9aGaa8xmaiaa=1cakmaadmaa baqcLbsacaWFXaGaa8xlaOWaaeWaaeaajugibiaa=fdacaWFRaGcda WcaaqaaKqzGeGaa8hUdiaa=HhakmaaCaaaleqajeaqbaqcLbmacaWF YoaaaaGcbaqcLbsacaWF4oGaa83kaiaa=fdaaaaakiaawIcacaGLPa aajugibiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTaGaa8hUdiaa =HhajuaGdaahaaqcbauabeaajugWaiaa=j7aaaaaaaGccaGLBbGaay zxaaWaaWbaaSqabKqaafaajugWaiaa=f7aaaaaaa@5AFA@

The coefficient of determination of Exponentiated Power Lindley Distribution is 0.975, which is higher than the coefficient of determination of PLD, GLD, Lindley Distribution, Exponentiated exponential, modified Weibull and Weibull distributions. The Exponentiated Power Lindley Distribution is good model for life time data.

The power Lindley distribution with its inference was proposed by Ghitany et al.22 and generalized by Liyanage.18

Ghitany et al.25 discussed the estimation of the reliability of a stress‒strength system from power Lindley distribution.

Inverse Lindley distribution

Sharma et al.26 investigated inverted version of the Lindley distribution that has upside down bathtub shaped failure rate. The inverse Lindley distribution (ILD) is defined by the following pdf

f(x;θ)= θ 2 1+θ ( 1+x x 3 ) e -θ x ; x,θ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=TdacaWF4oGaa8xkaiaa=1dakmaalaaa baqcLbsacaWF4oGcdaahaaWcbeqcbauaaKqzadGaa8NmaaaaaOqaaK qzGeGaa8xmaiaa=TcacaWF4oaaaOWaaeWaaeaadaWcaaqaaKqzGeGa a8xmaiaa=TcacaWF4baakeaajugibiaa=HhakmaaCaaaleqajeaqba qcLbmacaWFZaaaaaaaaOGaayjkaiaawMcaaKqzGeGaa8xzaOWaaWba aSqabKqaafaajuaGdaWcaaqcbauaaKqzadGaa8xlaiaa=H7aaKqaaf aajugWaiaa=HhaaaaaaKqzGeGaa8hiaiaa=TdacaWFGaGaa8hEaiaa =XcacaWF4oGaa8hiaiaa=5dacaWFWaaaaa@5EDA@

ILD is a mixture of inverse exponential distribution and special case of inverse gamma distribution with mixing constant θ 1+θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8hUdaGcbaqcLbsacaWFXaGaa83kaiaa=H7aaaaaaa@3D87@

The cdf of ILD is

F(x;θ,)=[ 1+ θ 1+θ 1 x ] e -θ x ;x,θ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=TdacaWF4oGaa8hlaiaa=LcacaWF9aGc daWadaqaaKqzGeGaa8xmaiaa=TcakmaalaaabaqcLbsacaWF4oaake aajugibiaa=fdacaWFRaGaa8hUdaaakmaalaaabaqcLbsacaWFXaaa keaajugibiaa=HhaaaaakiaawUfacaGLDbaajugibiaa=vgakmaaCa aaleqajeaqbaqcfa4aaSaaaKqaafaajugWaiaa=1cacaWF4oaajeaq baqcLbmacaWF4baaaaaajugibiaa=bcacaWF7aGaa8hEaiaa=Xcaca WF4oGaa8hiaiaa=5dacaWFWaaaaa@5ABC@

The hazard rate function is

h(x)= θ 2 (1+x) x 2 [ θ+x(1+θ)( e -θ x -1) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hU dOWaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFOaGaa8xmai aa=TcacaWF4bGaa8xkaaGcbaqcLbsacaWF4bGcdaahaaWcbeqcbaua aKqzadGaa8NmaaaakmaadmaabaqcLbsacaWF4oGaa83kaiaa=Hhaca WFOaGaa8xmaiaa=TcacaWF4oGaa8xkaiaa=HcacaWFLbGcdaahaaWc beqcbauaaKqbaoaalaaajeaqbaqcLbmacaWFTaGaa8hUdaqcbauaaK qzadGaa8hEaaaaaaqcLbsacaWFTaGaa8xmaiaa=LcaaOGaay5waiaa w2faaaaaaaa@5DEC@

The ILD fits quit well the data of survival of Head and Neck cancer patients. A new upside‒down bathtub‒shaped hazard rate model for survival data analysis was proposed by Sharma et al.27 by using transmuted Rayleigh distribution. Sharma et al.26 introduced the ILD as a one parameter model for a stress‒strength reliability model. Alkarni28 proposed three parameter ILD with application to maximum flood level data. Also29 studied the extension of ILD.

Truncated Lindleyd

Ahmed et al.30 introduced truncated version of Lindley Distribution. The truncated distribution is used where a random variable is restricted to be observed on some range. The truncated versions of the Lindley distribution, named as the upper truncated Lindley (UTL), lower truncated Lindley (LTL), double truncated Lindley (DTL) distributions are introduced.

The pdf of DTL distribution is denoted by g D (X; θ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=DgakmaaBaaajeaqbaqcLbmacaWFebaaleqaaKqzGeGaa8hkaiaa =HfacaWF7aGaa8hiaiaa=H7acaWFPaaaaa@41B7@ ,

g D (X;θ)= θ 2 1+θ (1+x) e -θx F(ζ;θ)-F(v;θ) ; 0vxζ< MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=DgakmaaBaaajeaqbaqcLbmacaWFebaaleqaaKqzGeGaa8hkaiaa =HfacaWF7aGaa8hUdiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hUdO WaaWbaaSqabKqaafaajugWaiaa=jdaaaaakeaajugibiaa=fdacaWF RaGaa8hUdaaakmaalaaabaqcLbsacaWFOaGaa8xmaiaa=TcacaWF4b Gaa8xkaiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTaGaa8hUdiaa =Hhaaaaakeaajugibiaa=zeacaWFOaaccaGae4NTdONaa83oaiaa=H 7acaWFPaGaa8xlaiaa=zeacaWFOaGaa8NDaiaa=TdacaWF4oGaa8xk aaaacaWFGaGaa83oaiaa=bcacaWFGaGaa8hmaiab+rMiJkaa=zhacq GFKjYOcaWF4bGae4hzImQae4NTdONaa8hpaiab+5HiLcaa@6D60@

At v = 0 and ζ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=z7a6jab=jziUkab=5HiLIqaaiaa+Xcaaaa@3EC6@  it reduces to baseline model here it is Lindly distribution, at v = 0, it gives the upper truncated distribution of the Lindley distribution and at ζ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaajugibi ab=z7a6jab=jziUkab=5HiLIqaaiaa+Xcaaaa@3EC6@  it is called the lower truncated distribution of the Lindly distribution.

The pdf of UTL distribution is,

g U (x;θ)= θ 2 (1+x) e -θ(x-ζ) (1+θ)( e θζ -1)-θζ ; 0xζ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=DgakmaaBaaajeaqbaqcLbmacaWFvbaaleqaaKqzGeGaa8hkaiaa =HhacaWF7aGaa8hUdiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hUdO WaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFOaGaa8xmaiaa =TcacaWF4bGaa8xkaiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTa Gaa8hUdiaa=HcacaWF4bGaa8xlaGGaaiab+z7a6jaa=Lcaaaaakeaa jugibiaa=HcacaWFXaGaa83kaiaa=H7acaWFPaGaa8hkaiaa=vgakm aaCaaaleqajeaqbaqcLbmacaWF4oGae4NTdOhaaKqzGeGaa8xlaiaa =fdacaWFPaGaa8xlaiaa=H7acqGF2oGEaaGaa8hiaiaa=TdacaWFGa Gaa8hiaiaa=bdacqGFKjYOcaWF4bGae4hzImQae4NTdOhaaa@6E0A@

when θ<1,  g U (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaakI 7acaGI8aGaaOymaiaakYcacaGIGaGaaO4zaOWaaSbaaKqaafaajugW aiaakwfaaSqabaqcLbsacaGIOaGaaOiEaiaakMcaaaa@43AD@  is uni‒modal and mode values is M 0 =(1-θ)/θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=1eakmaaBaaajeaqbaqcLbmacaWFWaaaleqaaKqzGeGaa8xpaiaa =HcacaWFXaGaa8xlaiaa=H7acaWFPaGaa83laiaa=H7aaaa@435D@  and the corresponding hazard rate function at t is

h(t)= θ 2 1+θ (1+t) e -θt F(ζ;θ)-F(t;θ) ; 0tζ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hDaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hU dOWaaWbaaSqabKqaafaajugWaiaa=jdaaaaakeaajugibiaa=fdaca WFRaGaa8hUdaaakmaalaaabaqcLbsacaWFOaGaa8xmaiaa=TcacaWF 0bGaa8xkaiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTaGaa8hUdi aa=rhaaaaakeaajugibiaa=zeacaWFOaaccaGae4NTdONaa83oaiaa =H7acaWFPaGaa8xlaiaa=zeacaWFOaGaa8hDaiaa=TdacaWF4oGaa8 xkaaaacaWFGaGaa83oaiaa=bcacaWFGaGaa8hmaiab+rMiJkaa=rha cqGFKjYOcqGF2oGEaaa@63AB@

The hazard rate function of UTL distribution is increasing in x and θ. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7acaWFUaaaaa@3AF3@

The pdf of LTL distribution is

g L (x;θ)= θ 2 (1+x) e -θ(x-v) 1+θ+θv; ; 0vx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=DgakmaaBaaajeaqbaqcLbmacaWFmbaaleqaaKqzGeGaa8hkaiaa =HhacaWF7aGaa8hUdiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hUdO WaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFOaGaa8xmaiaa =TcacaWF4bGaa8xkaiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTa Gaa8hUdiaa=HcacaWF4bGaa8xlaiaa=zhacaWFPaaaaaGcbaqcLbsa caWFXaGaa83kaiaa=H7acaWFRaGaa8hUdiaa=zhacaWF7aaaaiaa=b cacaWF7aGaa8hiaiaa=bcacaWFWaaccaGae4hzImQaa8NDaiab+rMi Jkaa=Hhaaaa@625E@

The truncated distributions can be quit effectively used to model the real problems and so we can use the truncated Lindley distributions in various fields including engineering, medical, finance and demography where such type of truncated data are commonly encountered. Among the three truncated versions DTL is more effective than UTL and LTL, also UTL is more effective than LTL.

Many researchers propose the truncated versions of the usual statistical distributions including, Ahmed et al.31 discussed the application of the truncated version of the Birnbaum‒Saunders (BS) distribution to improve a forecasting actuarial model and particularly. Zaninetti32 discussed the application of the truncated Pareto distribution to the statistical analysis of masses of stars and of diameters of asteroids for modelling data from insurance payments that establish a deductible. Zhang33 studied truncated Weibull distribution.

Discrete Lindley distribution

Deniz34 proposed the discrete Lindley distribution (DLD). It is obtained by discrediting the continuous failure model of the Lindley distribution. The probability mass function (pmf) of DLD is

f(x)=P(X=x)= λ x 1-logλ [λlogλ+(1-λ)(1-log λ x+1 )] ; x=0,1,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=LcacaWF9aGaa8huaiaa=HcacaWFybGa a8xpaiaa=HhacaWFPaGaa8xpaOWaaSaaaeaajugibiaa=T7akmaaCa aaleqajeaqbaqcLbmacaWF4baaaaGcbaqcLbsacaWFXaGaa8xlaiaa =XgacaWFVbGaa83zaiaa=T7aaaGaa83waiaa=T7acaWFSbGaa83Bai aa=DgacaWF7oGaa83kaiaa=HcacaWFXaGaa8xlaiaa=T7acaWFPaGa a8hkaiaa=fdacaWFTaGaa8hBaiaa=9gacaWFNbGaa83UdOWaaWbaaS qabKqaafaajugWaiaa=HhacaWFRaGaa8xmaaaajugibiaa=LcacaWF DbGaa8hiaiaa=TdacaWFGaGaa8hiaiaa=HhacaWF9aGaa8hmaiaa=X cacaWFXaGaa8hlaiaa=5cacaWFUaGaa8Nlaaaa@6CFF@

corresponding cdf is

F(x)= 1- λ x+1 +((2+x) λ x+1 -1)logλ 1-logλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xm aiaa=1cacaWF7oGcdaahaaWcbeqcbauaaKqzadGaa8hEaiaa=Tcaca WFXaaaaKqzGeGaa83kaiaa=HcacaWFOaGaa8Nmaiaa=TcacaWF4bGa a8xkaiaa=T7akmaaCaaaleqajeaqbaqcLbmacaWF4bGaa83kaiaa=f daaaqcLbsacaWFTaGaa8xmaiaa=LcacaWFSbGaa83Baiaa=DgacaWF 7oaakeaajugibiaa=fdacaWFTaGaa8hBaiaa=9gacaWFNbGaa83Uda aaaaa@5B95@

and the hazard rate function is

h(x)= λlogλ+(λ-1)(log λ x+1 -1 1-(1+x)logλ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa83U diaa=XgacaWFVbGaa83zaiaa=T7acaWFRaGaa8hkaiaa=T7acaWFTa Gaa8xmaiaa=LcacaWFOaGaa8hBaiaa=9gacaWFNbGaa83UdOWaaWba aSqabKqaafaajugWaiaa=HhacaWFRaGaa8xmaaaajugibiaa=1caca WFXaaakeaajugibiaa=fdacaWFTaGaa8hkaiaa=fdacaWFRaGaa8hE aiaa=LcacaWFSbGaa83Baiaa=DgacaWF7oaaaaaa@5BCE@

The mean and variance of DLD is increase with λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=T7aaaa@3A47@  and over‒dispersed, therefore, more flexible than the Poisson distribution to model actuarial data that commonly include the over‒dispersion phenomenon. It has an increasing hazard rate and it is unimodal. For large values of the parameter λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=T7aaaa@3A47@ , the mode moves to the right, showing a great versatility. Also if X follows a continuous Lindley distribution with parameter λ= e -θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=T7acaWF9aGaa8xzaOWaaWbaaSqabeaajugibiaa=1cacaWF4oaa aaaa@3E9B@ then the random variable Y= [X] follows a DLD with parameter λ= e -θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=T7acaWF9aGaa8xzaOWaaWbaaSqabeaajugibiaa=1cacaWF4oaa aaaa@3E9B@ , here [.] denotes the integer part.

The researchers who were discretise various continuous distributions including.35‒40

Poisson lindley distribution

A mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables. A compound distribution resembles in many ways the original distribution that generated it, but typically has greater variance, and often heavy tails as well. Any probability distribution on [0,) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaaiU facaaIWaGaaGilaiabg6HiLkaaiMcaaaa@3D78@  can function as the mixing distribution for a poisson mixture.

Discrete poisson lindley distribution

The discrete Poisson Lindley distribution (DPLD) is the Poisson distribution compounded with Lindley distribution and is proposed by Sankaran.41

The pdf of DPLD is,

f(x)=P(X=x)= θ 2 (x+θ+2) (θ+1) x+3 ; x=0,1,2..., θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=LcacaWF9aGaa8huaiaa=HcacaWFybGa a8xpaiaa=HhacaWFPaGaa8xpaOWaaSaaaeaajugibiaa=H7akmaaCa aaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa8hkaiaa=HhacaWFRaGa a8hUdiaa=TcacaWFYaGaa8xkaaGcbaqcLbsacaWFOaGaa8hUdiaa=T cacaWFXaGaa8xkaOWaaWbaaSqabKqaafaajugWaiaa=HhacaWFRaGa a83maaaaaaqcLbsacaWFGaGaa83oaiaa=bcacaWFGaGaa8hEaiaa=1 dacaWFWaGaa8hlaiaa=fdacaWFSaGaa8Nmaiaa=5cacaWFUaGaa8Nl aiaa=XcacaWFGaGaa8hiaiaa=H7acaWF+aGaa8hmaaaa@645F@

The DPLD is over dispersed, so it can be used in fields like biological science and medical science. In the field of genetics and ecology DPLD gives much closer fit than Poisson distribution and thus it can be considered as an important tool for modeling data in these fields.

Ghitany42 discussed the estimation methods for the DPLD and its applications. The DPLD has been generalized by many researchers. Shanker43 studied a two parameter Poisson‒Lindley distribution by compounding Poisson distribution with a two parameter Lindley distribution. Shanker44 obtained a new quasi Poisson‒Lindley distribution by compounding Poisson distribution with a new quasi Lindley distribution introduced by Shanker et al.45 obtained a discrete two parameter Poisson Lindley distribution by mixing Poisson distribution with a two parameter Lindley distribution. Also Shanker46 studied Poisson‒Lindley distribution and its application to biological science.

Discrete poisson‒akash distribution (DPAD)

Shanker47 proposed DPAD. The pdf of Poisson mixture of Akash distribution is

f(x)= θ 3 θ 2 +2 x 2 +3x+( θ 2 +2θ+3) (θ+1) x+3 ; x=0,1,2...,θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hU dOWaaWbaaSqabKqaafaajugWaiaa=ndaaaaakeaajugibiaa=H7akm aaCaaaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa83kaiaa=jdaaaGc daWcaaqaaKqzGeGaa8hEaOWaaWbaaSqabKqaafaajugWaiaa=jdaaa qcLbsacaWFRaGaa83maiaa=HhacaWFRaGaa8hkaiaa=H7akmaaCaaa leqajeaqbaqcLbmacaWFYaaaaKqzGeGaa83kaiaa=jdacaWF4oGaa8 3kaiaa=ndacaWFPaaakeaajugibiaa=HcacaWF4oGaa83kaiaa=fda caWFPaGcdaahaaWcbeqcbauaaKqzadGaa8hEaiaa=TcacaWFZaaaaa aajugibiaa=bcacaWF7aGaa8hiaiaa=bcacaWF4bGaa8xpaiaa=bda caWFSaGaa8xmaiaa=XcacaWFYaGaa8Nlaiaa=5cacaWFUaGaa8hlai aa=H7acaWF+aGaa8hmaaaa@6EC2@

DPAD has an increasing hazard rate and unimodal and always over‒dispersed thus it is a suitable model for count data which are over‒dispersed. DPAD gives much closer fit than Poisson distribution and DPLD in almost all cases, DPAD has some flexibility over DPLD.

Shanker et al.48 discussed Poisson‒Akash Distribution and its Applications. Shanker49 has also introduced size based and zero truncated version of Poisson Akash distribution and studied their properties.

Transmuted lindley distribution

Merovci50 Proposed Transmuted Lindley Distribution. The pdf of Transmuted Lindley Distribution is,

f(x)= θ 2 1+θ (1+x) e -θx ( 1-λ+2λ θ+1+θx θ+1 e -θx ) ; x>0, θ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hU dOWaaWbaaSqabKqaafaajugWaiaa=jdaaaaakeaajugibiaa=fdaca WFRaGaa8hUdaaacaWFOaGaa8xmaiaa=TcacaWF4bGaa8xkaiaa=vga kmaaCaaaleqajeaqbaqcLbmacaWFTaGaa8hUdiaa=HhaaaGcdaqada qaaKqzGeGaa8xmaiaa=1cacaWF7oGaa83kaiaa=jdacaWF7oGcdaWc aaqaaKqzGeGaa8hUdiaa=TcacaWFXaGaa83kaiaa=H7acaWF4baake aajugibiaa=H7acaWFRaGaa8xmaaaacaWFLbGcdaahaaWcbeqcbaua aKqzadGaa8xlaiaa=H7acaWF4baaaaGccaGLOaGaayzkaaqcLbsaca WFGaGaa83oaiaa=bcacaWFGaGaa8hEaiaa=5dacaWFWaGaa8hlaiaa =bcacaWF4oGaa8Npaiaa=bdaaaa@6D5E@

The transmuted Lindley distribution is an extended model to analyze more complex data and it generalizes some of the widely used distributions, at λ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBca aI9aGaaGimaaaa@3BA5@  it gives Lindley distribution. The corresponding cdf is,

F(x)=( 1- θ+1+θx θ+1 e -θx )( 1+ θ+1+θx θ+1 e -θx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=zeacaWFOaGaa8hEaiaa=LcacaWF9aGcdaqadaqaaKqzGeGaa8xm aiaa=1cakmaalaaabaqcLbsacaWF4oGaa83kaiaa=fdacaWFRaGaa8 hUdiaa=HhaaOqaaKqzGeGaa8hUdiaa=TcacaWFXaaaaiaa=vgakmaa CaaaleqajeaqbaqcLbmacaWFTaGaa8hUdiaa=HhaaaaakiaawIcaca GLPaaadaqadaqaaKqzGeGaa8xmaiaa=TcakmaalaaabaqcLbsacaWF 4oGaa83kaiaa=fdacaWFRaGaa8hUdiaa=HhaaOqaaKqzGeGaa8hUdi aa=TcacaWFXaaaaiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTaGa a8hUdiaa=HhaaaaakiaawIcacaGLPaaaaaa@6183@

The hazard rate function is

h(x)= θ 2 (1+x)( 1-λ+2λ θ+1+θx θ+1 e -θx ) (1+θ+θx)( λ-1-λ θ+1+θx θ+1 e -θx ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFOaGaa8hEaiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8hU dOWaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFOaGaa8xmai aa=TcacaWF4bGaa8xkaOWaaeWaaeaajugibiaa=fdacaWFTaGaa83U diaa=TcacaWFYaGaa83UdOWaaSaaaeaajugibiaa=H7acaWFRaGaa8 xmaiaa=TcacaWF4oGaa8hEaaGcbaqcLbsacaWF4oGaa83kaiaa=fda aaGaa8xzaOWaaWbaaSqabKqaafaajugWaiaa=1cacaWF4oGaa8hEaa aaaOGaayjkaiaawMcaaaqaaKqzGeGaa8hkaiaa=fdacaWFRaGaa8hU diaa=TcacaWF4oGaa8hEaiaa=LcakmaabmaabaqcLbsacaWF7oGaa8 xlaiaa=fdacaWFTaGaa83UdOWaaSaaaeaajugibiaa=H7acaWFRaGa a8xmaiaa=TcacaWF4oGaa8hEaaGcbaqcLbsacaWF4oGaa83kaiaa=f daaaGaa8xzaOWaaWbaaSqabKqaafaajugWaiaa=1cacaWF4oGaa8hE aaaaaOGaayjkaiaawMcaaaaaaaa@7897@

From this if λ=1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBca aI9aGaaGymaiaaiYcaaaa@3C5C@  the hazard rate is decreasing. Aryal51 studied Transmuted Weibull distribution and its properties.

Wrapped lindley distribution

A wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n‒sphere. The cases of wrapped Lindley distribution have been studied extensively by Joshi.52 The wrapped Lindley (WL) random variable is defined as θ=X(mod2π) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7acaWF9aGaa8hwaiaa=HcacaWFTbGaa83Baiaa=rgacaWFYaGa a8hWdiaa=Lcaaaa@41E8@ , such that for θÎ[0,2π] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaacaWF4o Gaa8NZaiaa=TfacaWFWaGaa8hlaiaa=jdacaWFapGaa8xxaaaa@4013@ , the pdf is given by

g(θ)= λ 2 1+λ e -λθ [ 1+θ 1- e -2πλ + 2π e -2πλ (1- e -2πλ ) 2 ],θ[0,2π),λ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=DgacaWFOaGaa8hUdiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa83U dOWaaWbaaSqabKqaafaajugWaiaa=jdaaaaakeaajugibiaa=fdaca WFRaGaa83UdaaacaWFLbGcdaahaaWcbeqcbauaaKqzadGaa8xlaiaa =T7acaWF4oaaaOWaamWaaeaadaWcaaqaaKqzGeGaa8xmaiaa=Tcaca WF4oaakeaajugibiaa=fdacaWFTaGaa8xzaOWaaWbaaSqabKqaafaa jugWaiaa=1cacaWFYaGaa8hWdiaa=T7aaaaaaKqzGeGaa83kaOWaaS aaaeaajugibiaa=jdacaWFapGaa8xzaOWaaWbaaSqabKqaafaajugW aiaa=1cacaWFYaGaa8hWdiaa=T7aaaaakeaajugibiaa=HcacaWFXa Gaa8xlaiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTaGaa8Nmaiaa =b8acaWF7oaaaKqzGeGaa8xkaOWaaWbaaSqabKqaafaajugWaiaa=j daaaaaaaGccaGLBbGaayzxaaqcLbsacaaMi8Uaa8hlaiaaysW7caWF 4oaccaGae4hcI4Saa83waiaa=bdacaWFSaGaa8Nmaiaa=b8acaWFPa Gaa8hlaiaa=T7acaWF+aGaa8hmaaaa@7E89@

The random variable θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7aaaa@3A44@  having wrapped Lindley distribution is denoted by θ:WL(λ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7acaWF6aGaa83vaiaa=XeacaWFOaGaa83Udiaa=Lcaaaa@3F36@ . The cdf of WL is

G(θ)= 1 1- e -2πλ ( 1- e -λθ - λθ 1+λ e λθ )- 2πλ 1+λ ( 1- e λθ )( e -2πλ 1- e -2πλ )θ[0,2π),λ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=DeacaWFOaGaa8hUdiaa=LcacaWF9aGcdaWcaaqaaKqzGeGaa8xm aaGcbaqcLbsacaWFXaGaa8xlaiaa=vgakmaaCaaaleqajeaqbaqcLb macaWFTaGaa8Nmaiaa=b8acaWF7oaaaaaakmaabmaabaqcLbsacaWF XaGaa8xlaiaa=vgakmaaCaaaleqajeaqbaqcLbmacaWFTaGaa83Udi aa=H7aaaqcLbsacaWFTaGcdaWcaaqaaKqzGeGaa83Udiaa=H7aaOqa aKqzGeGaa8xmaiaa=TcacaWF7oaaaiaa=vgakmaaCaaaleqajeaqba qcLbmacaWF7oGaa8hUdaaaaOGaayjkaiaawMcaaKqzGeGaa8xlaOWa aSaaaeaajugibiaa=jdacaWFapGaa83UdaGcbaqcLbsacaWFXaGaa8 3kaiaa=T7aaaGcdaqadaqaaKqzGeGaa8xmaiaa=1cacaWFLbGcdaah aaWcbeqcbauaaKqzadGaa83Udiaa=H7aaaaakiaawIcacaGLPaaada qadaqaamaalaaabaqcLbsacaWFLbGcdaahaaWcbeqcbauaaKqzadGa a8xlaiaa=jdacaWFapGaa83UdaaaaOqaaKqzGeGaa8xmaiaa=1caca WFLbGcdaahaaWcbeqcbauaaKqzadGaa8xlaiaa=jdacaWFapGaa83U daaaaaaakiaawIcacaGLPaaajugibiaa=H7aiiaacqGFiiIZcaWFBb Gaa8hmaiaa=XcacaWFYaGaa8hWdiaa=LcacaWFSaGaa83Udiaa=5da caWFWaaaaa@8A6E@

Joshi52 showed that wrapped Lindley distribution give good fit to the data set (orientations of 76 turtles after laying eggs and is given in Table 1 (Rao53) than wrapped exponential distribution.

Measure

Exponential distribution

Lindley distribution

coefficient of variation MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajugibiaado gacaWGVbGaamyzaiaadAgacaWGMbGaamyAaiaadogacaWGPbGaamyz aiaad6gacaWG0bGaaGiiaiaad+gacaWGMbGaaGiiaiaadAhacaWGHb GaamOCaiaadMgacaWGHbGaamiDaiaadMgacaWGVbGaamOBaaaa@4EE2@

1

θ 2 +4θ+2 θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaGcaaqaam aalaaabaqcLbsacqaH4oqCkmaaCaaaleqajeaqbaqcLbmacaaIYaaa aKqzGeGaey4kaSIaaGinaiabeI7aXjabgUcaRiaaikdaaOqaaKqzGe GaeqiUdeNaey4kaSIaaGOmaaaaaSqabaaaaa@46BB@

coefficient of Skewness( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=ngacaWFVbGaa8xzaiaa=zgacaWFMbGaa8xAaiaa=ngacaWFPbGa a8xzaiaa=5gacaWF0bGaa8hiaiaa=9gacaWFMbGaa8hiaiaa=nfaca WFRbGaa8xzaiaa=DhacaWFUbGaa8xzaiaa=nhacaWFZbGaa8hkaOWa aOaaaeaajugibiaa=j7akmaaBaaaleaajugibiaa=fdaaSqabaaabe aajugibiaa=Lcaaaa@52C9@

2

2( θ 3 +6 θ 2 +6θ+2) ( θ 2 +4θ+2) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8Nmaiaa=HcacaWF4oGcdaahaaWcbeqcbauaaKqzadGa a83maaaajugibiaa=TcacaWF2aGaa8hUdOWaaWbaaSqabKqaafaaju gWaiaa=jdaaaqcLbsacaWFRaGaa8Nnaiaa=H7acaWFRaGaa8Nmaiaa =LcaaOqaaKqzGeGaa8hkaiaa=H7akmaaCaaaleqajeaqbaqcLbmaca WFYaaaaKqzGeGaa83kaiaa=rdacaWF4oGaa83kaiaa=jdacaWFPaGc daahaaWcbeqcbauaaKqzadGaa83maiaa=9cacaWFYaaaaaaaaaa@56B5@

coefficient of Kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=ngacaWFVbGaa8xzaiaa=zgacaWFMbGaa8xAaiaa=ngacaWFPbGa a8xzaiaa=5gacaWF0bGaa8hiaiaa=9gacaWFMbGaa8hiaiaa=Teaca WF1bGaa8NCaiaa=rhacaWFVbGaa83Caiaa=LgacaWFZbGaa8hiaiaa =HcacaWFYoGcdaWgaaqcbauaaKqzadGaa8NmaaWcbeaajugibiaa=L caaaa@53B1@

9

3(3 θ 4 +24 θ 3 +44 θ 2 +32θ+8) ( θ 2 +4θ+2) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa83maiaa=HcacaWFZaGaa8hUdOWaaWbaaSqabKqaafaa jugWaiaa=rdaaaqcLbsacaWFRaGaa8Nmaiaa=rdacaWF4oGcdaahaa WcbeqcbauaaKqzadGaa83maaaajugibiaa=TcacaWF0aGaa8hnaiaa =H7akmaaCaaaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa83kaiaa=n dacaWFYaGaa8hUdiaa=TcacaWF4aGaa8xkaaGcbaqcLbsacaWFOaGa a8hUdOWaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsacaWFRaGaa8 hnaiaa=H7acaWFRaGaa8Nmaiaa=LcakmaaCaaaleqajeaqbaqcLbma caWFYaaaaaaaaaa@5DB2@

Index of dispersion, γ= μ 2 μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=LeacaWFUbGaa8hzaiaa=vgacaWF4bGaa8hiaiaa=9gacaWFMbGa a8hiaiaa=rgacaWFPbGaa83Caiaa=bhacaWFLbGaa8NCaiaa=nhaca WFPbGaa83Baiaa=5gacaWFSaGaa8hiaiaa=n7acaWF9aGcdaWcaaqa aKqzGeGaa8hVdOWaaSbaaKqaafaajugWaiaa=jdaaSqabaaakeaaju gibiaa=X7akmaaBaaajeaqbaqcLbmacaWFXaaaleqaaKqzadGaa83j aaaaaaa@5784@

1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8xmaaGcbaqcLbsacaWF4oaaaaaa@3B9F@

θ 2 +4θ+2 ) 2 θ( θ 2 +3θ+2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8hUdOWaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsa caWFRaGaa8hnaiaa=H7acaWFRaGaa8Nmaiaa=LcakmaaCaaaleqaje aqbaqcLbmacaWFYaaaaaGcbaqcLbsacaWF4oGaa8hkaiaa=H7akmaa CaaaleqajeaqbaqcLbmacaWFYaaaaKqzGeGaa83kaiaa=ndacaWF4o Gaa83kaiaa=jdacaWFPaaaaaaa@4F9D@

hazard rate function MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=HgacaWFHbGaa8NEaiaa=fgacaWFYbGaa8hzaiaa=bcacaWFYbGa a8xyaiaa=rhacaWFLbGaa8hiaiaa=zgacaWF1bGaa8NBaiaa=ngaca WF0bGaa8xAaiaa=9gacaWFUbaaaa@4AE8@

θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=H7aaaa@3A44@

θ 2 (1+x) 1+θ+θx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8hUdOWaaWbaaSqabKqaafaajugWaiaa=jdaaaqcLbsa caWFOaGaa8xmaiaa=TcacaWF4bGaa8xkaaGcbaqcLbsacaWFXaGaa8 3kaiaa=H7acaWFRaGaa8hUdiaa=Hhaaaaaaa@4703@

mean residual life function (m(x)) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaajugibi aa=1gacaWFLbGaa8xyaiaa=5gacaWFGaGaa8NCaiaa=vgacaWFZbGa a8xAaiaa=rgacaWF1bGaa8xyaiaa=XgacaWFGaGaa8hBaiaa=Lgaca WFMbGaa8xzaiaa=bcacaWFMbGaa8xDaiaa=5gacaWFJbGaa8hDaiaa =LgacaWFVbGaa8NBaiaa=bcacaWFOaGaa8xBaiaa=HcacaWF4bGaa8 xkaiaa=Lcaaaa@5631@

1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8xmaaGcbaqcLbsacaWF4oaaaaaa@3B9F@

θ+2+θx θ(1+θ+θx) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9 Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaG qaaKqzGeGaa8hUdiaa=TcacaWFYaGaa83kaiaa=H7acaWF4baakeaa jugibiaa=H7acaWFOaGaa8xmaiaa=TcacaWF4oGaa83kaiaa=H7aca WF4bGaa8xkaaaaaaa@4737@

Table 1 Different measures of exponential and Lindley distributions

Conclusion

This paper studied the well‒established and widely used Lindley distribution and its generalizations. The variety of generalizations or parameters induction can be used to handle various real data sets with complex structure. These models will be useful for constructing probability models and may help the development of new classes from the Lindley distribution in future.

Acknowledgement

None.

Conflict of interest statement

The author declares that there in none of the conflicts.

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