ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 6 - 2016
On Modeling of Lifetime Data Using Akash, Shanker, Lindley and Exponential Distributions
Rama Shanker1* and Hagos Fesshaye2
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
Received:May 03, 2016 | Published: June 24, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea; Email:
Citation: Shanker R, Fesshaye H (2016) On Modeling of Lifetime Data Using Akash, Shanker, Lindley and Exponential Distributions. Biom Biostat Int J 3(6): 00084. DOI: 10.15406/bbij.2016.03.00084

Abstract

The statistical analysis and modeling of lifetime data are crucial for statisticians and research workers in almost all applied sciences including engineering, biomedical science, insurance, and finance, amongst others. The two important and popular one parameter distributions for modeling lifetime data are exponential and Lindley distributions. Shanker et al. [1] observed that there are many lifetime data where these distributions are not suitable from theoretical and applied point of view. Recently Shanker [2,3] has introduced two one parameter Lifetime distribution namely “Akash distribution” and “Shanker distribution” for modeling lifetime data.

In the present paper the relationships and comparative studies of Akash, Shanker, Lindley and exponential distributions, their distributional properties and estimation of parameter have been discussed. The applications, goodness of fit and theoretical justifications of these distributions for modeling life time data through various examples from engineering, medical science and other fields have been discussed and explained.

Keywords: Akash distribution; Shanker distribution; Lindley distribution; Exponential distribution; Statistical properties; Estimation of parameter; Goodness of fit

Introduction

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

In statistics literature a number of lifetime distributions for modeling lifetime data-sets have been proposed. In this paper, the main objective is to have a critical and comparative study on one parameter lifetime distributions namely, Akash, Shanker, Lindley and exponential and their applications for modeling lifetime dats-sets from engineering, medical sciences, and other fields of knowledge.

Akash, Shanker, Lindley and Exponential Distributions

Akash distribution introduced by Shanker [2] for modeling lifetime data from engineering and medical science is a two-component mixture of an exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@ distribution and a gamma ( 3,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIZaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B30@ distribution with their mixing proportions θ 2 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaaaaaa@3ED2@  and 2 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaIYaaaaaaa@3C3E@ respectively. Shanker [2] has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability , amongst others. Shanker et al. [3] has detailed study about modeling of various lifetime data from different fields using Akash, Lindley and exponential distributions and concluded that Akash distribution gives better fit in most of the lifetime data. Shanker [5] has also obtained a Poisson mixture of Akash distribution named, “Poisson-Akash (PAD)” for modeling count data.

Shanker distribution introduced by Shanker [2] for modeling lifetime data from engineering and medical science is a two- component mixture of an exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@  distribution and a gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@ distribution with their mixing proportions θ 2 θ 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaaaaaa@3ED1@  and 1 θ 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIXaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaIXaaaaaaa@3C3C@ respectively. Shanker [3] has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability , amongst others. Shanker [6] has also obtained a Poisson mixture of Shanker distribution named, “Poisson-Shanker (PSD)” for modeling count data.

Lindley [7] distribution is a two-component mixture of an exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@  distribution and a gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@ distribution with their mixing proportions θ θ+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCaeaacqaH4oqCcqGHRaWkcaaIXaaaaaaa@3B9D@  and 1 θ+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIXaaabaGaeqiUdeNaey4kaSIaaGymaaaaaaa@3AA2@ respectively. A detailed study about its various mathematical properties, estimation of parameter and application showing the superiority of Lindley distribution over exponential distribution for the waiting times before service of the bank customers has been done by Ghitany et al. [8]. A number of researchers have studied in detail the generalized, extended, mixtures and modified forms of Lindley distribution including Sankaran [9], Zakerzadeh & Dolati [10], Nadarajah et al. [11], Bakouch et al. [12], Shanker & Mishra [13,14], Shanker & Amanuel [15], Shanker et al. [16,17], Ghitany et al. [18], are some among others.

In statistical literature, exponential distribution was the first widely used lifetime model in areas ranging from studies on the lifetimes of manufactured to research involving survival or remission times in chronic diseases. The main reason for its wide usefulness and applicability as lifetime model is partly because of the availability of simple statistical methods for it and partly because it appeared to be suitable for representing the lifetimes of many things such as various types of manufactured items.

Let T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamivaa aa@375D@ be a continuous random variable representing the lifetimes of individuals in some population The expressions for probability density function, f( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@39F1@ , cumulative distribution function, F( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@39D1@ ,  hazard rate function, h( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@39F3@ , mean residual life function, m( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiDaaGaayjkaiaawMcaaaaa@39F8@ , mean μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aaaa@3CDC@ , variance μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@ , coefficient of variation (C.V.), coefficient of Skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@ , coefficient of Kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B47@ , and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@  of Akash and Shanker distributions introduced by Shanker [2,3] are summarized in Table 1 and that of Lindley and exponential distributions are in Table 2.

A table of values for coefficient of variation (C.V.), coefficient of Skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aadaGcaaqaaiabek7aInaaBaaajuaibaGaaGymaaqcfayabaaabeaa aiaawIcacaGLPaaaaaa@3B56@ , coefficient of Kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHYoGydaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMca aaaa@3B47@ , and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaHZoWzaiaawIcacaGLPaaaaaa@39B4@  for Akash , Shanker and Lindley distributions for varying values of their parameter are summarized in the Table 3.

The conditions under which Akash, Shanker and Lindley distributions are over-dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E24@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E26@  , and under-dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH+aGpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E28@  are summarized in Table 4.

The graphs of C.V, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@  of Akash, Shanker and Lindley distributions for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  are shown in Figure 1.

Parameter Estimation

Estimation of the parameter of Akash distribution

Assuming ( t 1 , t 2 , t 3 ,..., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG0bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaaGPaVlaa dshadaWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaaMc8UaamiDam aaBaaajuaibaGaaG4maaqcfayabaGaaiilaiaaykW7caaMc8UaaiOl aiaac6cacaGGUaGaaGPaVlaaykW7caGGSaGaamiDamaaBaaajuaiba GaamOBaaqcfayabaaacaGLOaGaayzkaaaaaa@50A4@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  from Akash distribution, the maximum likelihood estimate (MLE) θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  and the method moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of following cubic equation.

t ¯ θ 3 θ 2 +2 t ¯ θ6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiDay aaraGaeqiUde3aaWbaaeqajuaibaGaaG4maaaajuaGcqGHsislcqaH 4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdaceWG0b GbaebacqaH4oqCcqGHsislcaaI2aGaeyypa0JaaGimaaaa@46F5@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiDay aaraaaaa@3795@ is the sample mean                                                       

Estimation of the parameter of Shanker distribution                              

Let ( t 1 , t 2 , t 3 ,..., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG0bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaaGPaVlaa dshadaWgaaqcfasaaiaaikdaaKqbagqaaiaacYcacaaMc8UaamiDam aaBaaajuaibaGaaG4maaqcfayabaGaaiilaiaaykW7caaMc8UaaiOl aiaac6cacaGGUaGaaGPaVlaaykW7caGGSaGaamiDamaaBaaajuaiba GaamOBaaqcfayabaaacaGLOaGaayzkaaaaaa@50A4@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@  from Shanker distribution. The maximum likelihood estimate (MLE) θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of the following non-linear equation.

2n θ( θ 2 +1 ) + i=1 n 1 θ+ t i n t ¯ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaGaamOBaaqaaiabeI7aXnaabmaabaGaeqiUde3aaWbaaeqa juaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaai abgUcaRmaaqahabaWaaSaaaeaacaaIXaaabaGaeqiUdeNaey4kaSIa amiDamaaBaaajuaibaGaamyAaaqcfayabaaaaaqcfasaaiaadMgacq GH9aqpcaaIXaaabaGaamOBaaqcfaOaeyyeIuoacqGHsislcaWGUbGa aGPaVlqadshagaqeaiabg2da9iaaicdaaaa@543C@                                         

The method of moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is the solution of the following cubic equation

t ¯ θ 3 θ 2 + t ¯ θ2=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiDay aaraGaaGPaVlabeI7aXnaaCaaabeqcfasaaiaaiodaaaqcfaOaeyOe I0IaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkceWG0b GbaebacaaMc8UaeqiUdeNaeyOeI0IaaGOmaiabg2da9iaaicdaaaa@494B@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiDay aaraaaaa@3795@ is the sample mean.                   

Estimation of the parameter of Lindley distribution

Assuming  ( t 1 , t 2 ,...., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG0bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiDamaa BaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlai aac6cacaGGSaGaamiDamaaBaaajuaibaGaamOBaaqcfayabaaacaGL OaGaayzkaaaaaa@44D1@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@  from Lindley distribution, the maximum likelihood estimate (MLE) θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  and the method moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is given by

θ ^ = ( t ¯ 1 )+ ( t ¯ 1 ) 2 +8 t ¯ 2 t ¯ ; t ¯ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpdaWcaaqaaiabgkHiTmaabmaabaGabmiDayaaraGa eyOeI0IaaGymaaGaayjkaiaawMcaaiabgUcaRmaakaaabaWaaeWaae aaceWG0bGbaebacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaeqa juaibaGaaGOmaaaajuaGcqGHRaWkcaaI4aGaaGPaVlqadshagaqeaa qabaaabaGaaGOmaiaaykW7ceWG0bGbaebaaaGaai4oaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlqadshagaqeaiabg6da+iaaic daaaa@59C9@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiDay aaraaaaa@3795@ is the sample mean.

Estimation of the parameter of Exponential distribution

Assuming ( t 1 , t 2 ,...., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWG0bWaaSbaaKqbGeaacaaIXaaajuaGbeaacaGGSaGaamiDamaa BaaajuaibaGaaGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlai aac6cacaGGSaGaamiDamaaBaaajuaibaGaamOBaaqcfayabaaacaGL OaGaayzkaaaaaa@44D1@  be a random sample of size n from exponential distribution, the maximum likelihood estimate (MLE) θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaaaaa@384A@  and the method moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaGaaaaa@3849@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  is is given by θ ^ = 1 t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOafqiUde NbaKaacqGH9aqpdaWcaaqaaiaaigdaaeaaceWG0bGbaebaaaaaaa@3B2C@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOabmiDay aaraaaaa@3795@ is the sample mean.

Applications and Goodness of Fit

In this section the goodness of fit test of Akash, Shanker, Lindley and exponential distributions for following sixteen real lifetime data- sets using maximum likelihood estimate have been discussed.

In order to compare the goodness of fit of Akash, Shanker, Lindley and exponential distributions, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics ( Kolmogorov-Smirnov Statistics)  for all sixteen real lifetime data- sets have been computed and presented in Table 5.  The formulae for computing AIC, AICC, BIC, and K-S Statistics are as follows:

AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaaikdacaWGRbaaaa@40D2@ , AICC=AIC+ 2k( k+1 ) ( nk1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyqai aadMeacaWGdbGaam4qaiabg2da9iaadgeacaWGjbGaam4qaiabgUca RmaalaaabaGaaGOmaiaadUgadaqadaqaaiaadUgacqGHRaWkcaaIXa aacaGLOaGaayzkaaaabaWaaeWaaeaacaWGUbGaeyOeI0Iaam4Aaiab gkHiTiaaigdaaiaawIcacaGLPaaaaaaaaa@49BF@ , BIC=2lnL+klnn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOqai aadMeacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamit aiabgUcaRiaadUgaciGGSbGaaiOBaiaad6gaaaa@42EE@ and D= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamirai abg2da9maaxababaGaae4uaiaabwhacaqGWbaajuaibaGaamiEaaqc fayabaWaaqWaaeaacaWGgbWaaSbaaKqbGeaacaWGUbaajuaGbeaada qadaqaaiaadIhaaiaawIcacaGLPaaacqGHsislcaWGgbWaaSbaaeaa caaIWaaabeaadaqadaqaaiaadIhaaiaawIcacaGLPaaaaiaawEa7ca GLiWoaaaa@4A57@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4Aaa aa@3774@ = the number of parameters, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOBaa aa@3777@ = the sample size and F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamOBaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaaaaa@3BA5@ is the empirical distribution function. The best distribution is the distribution which corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeyOeI0 IaaGOmaiGacYgacaGGUbGaamitaaaa@3AE2@ , AIC, AICC, BIC, and K-S statistics.

The best fitting has been shown by making -2ln L, AIC, AICC, BIC, and K-S Statistics in bold.

Conclusions

In this paper an attempt has been made to find the suitability of Akash, Shanker, Lindley and exponential distributions for modeling real lifetime data from engineering, medical science and other fields of knowledge. A table for values of the various characteristics of Akash, Shanker, and Lindley distributions has been presented for varying values of their parameter which reflects their nature and behavior. The conditions under which Akash, shanker, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed have been given. The goodness of fit test of Akash, Shanker, Lindley and exponential distributions for sixteen real lifetime data-sets have been presented using Kolmogorov-Smirnov test to test their suitability for modeling lifetime data.

Akash Distribution

Shanker Distribution

f( t )= θ 3 θ 2 +2 ( 1+ t 2 ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaG4maaaaaKqbagaacqaH4oqCdaahaaqabK qbGeaacaaIYaaaaKqbakabgUcaRiaaikdaaaWaaeWaaeaacaaIXaGa ey4kaSIaamiDamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLOaGaay zkaaGaamyzamaaCaaabeqcfasaaiabgkHiTiabeI7aXjaaykW7caWG 0baaaaaa@4F60@

f( t )= θ 2 θ 2 +1 ( θ+t ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaGOmaaaaaKqbagaacqaH4oqCdaahaaqabK qbGeaacaaIYaaaaKqbakabgUcaRiaaigdaaaWaaeWaaeaacqaH4oqC cqGHRaWkcaWG0baacaGLOaGaayzkaaGaamyzamaaCaaabeqcfasaai abgkHiTiabeI7aXjaaykW7caWG0baaaKqbakaaykW7aaa@50D8@

F( t )=1[ 1+ θt( θt+2 ) θ 2 +2 ] e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaaykW7caWG0b WaaeWaaeaacqaH4oqCcaaMc8UaamiDaiabgUcaRiaaikdaaiaawIca caGLPaaaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgU caRiaaikdaaaaacaGLBbGaayzxaaGaamyzamaaCaaabeqcfasaaiab gkHiTiabeI7aXjaaykW7caWG0baaaaaa@5708@

F( t )=1[ 1+ θt θ 2 +1 ] e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWadaqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaaykW7caWG0b aabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI XaaaaaGaay5waiaaw2faaiaadwgadaahaaqabKqbGeaacqGHsislcq aH4oqCcaaMc8UaamiDaaaaaaa@4FA6@

h( t )= θ 3 ( 1+ t 2 ) θt( θt+2 )+( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaG4maaaajuaGdaqadaqaaiaaigdacqGHRa WkcaWG0bWaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawIcacaGLPaaa aeaacqaH4oqCcaaMc8UaamiDamaabmaabaGaeqiUdeNaaGPaVlaads hacqGHRaWkcaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWaaeaacqaH 4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdaaiaawI cacaGLPaaaaaaaaa@5707@

h( t )= θ 2 ( θ+t ) ( θ 2 +1 )+θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiabeI7aXjabgU caRiaadshaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7aXnaaCaaa beqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaGaayjkaiaawMcaai abgUcaRiabeI7aXjaaykW7caWG0baaaaaa@4F05@

m( t )= θ 2 t 2 +4θt+( θ 2 +6 ) θ[ θt( θt+2 )+( θ 2 +2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaGOmaaaajuaGcaaMc8UaamiDamaaCaaabe qcfasaaiaaikdaaaqcfaOaey4kaSIaaGinaiabeI7aXjaaykW7caWG 0bGaey4kaSYaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaK qbakabgUcaRiaaiAdaaiaawIcacaGLPaaaaeaacqaH4oqCdaWadaqa aiabeI7aXjaaykW7caWG0bWaaeWaaeaacqaH4oqCcaaMc8UaamiDai abgUcaRiaaikdaaiaawIcacaGLPaaacqGHRaWkdaqadaqaaiabeI7a XnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaGaayjkai aawMcaaaGaay5waiaaw2faaaaaaaa@664F@

m( t )= θ 2 +θt+2 θ( θ 2 +θt+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcqaH4oqCcaaMc8 UaamiDaiabgUcaRiaaikdaaeaacqaH4oqCdaqadaqaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaeqiUdeNaaGPaVlaads hacqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaaaa@5260@

μ 1 = θ 2 +6 θ( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaju aGcqGHRaWkcaaI2aaabaGaeqiUde3aaeWaaeaacqaH4oqCdaahaaqa bKqbGeaacaaIYaaaaKqbakabgUcaRiaaikdaaiaawIcacaGLPaaaaa aaaa@4B11@

μ 1 = θ 2 +2 θ( θ 2 +1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaaju aGcqGHRaWkcaaIYaaabaGaeqiUde3aaeWaaeaacqaH4oqCdaahaaqa bKqbGeaacaaIYaaaaKqbakabgUcaRiaaigdaaiaawIcacaGLPaaaaa aaaa@4B0C@

μ 2 = θ 4 +16 θ 2 +12 θ 2 ( θ 2 +2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaGymaiaaiAdacq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaigdacaaI YaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaai abeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaGa ayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@5114@

μ 2 = θ 4 +4 θ 2 +2 θ 2 ( θ 2 +1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaGinaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaqaaiabeI7a XnaaCaaabeqcfasaaiaaikdaaaqcfa4aaeWaaeaacqaH4oqCdaahaa qabKqbGeaacaaIYaaaaKqbakabgUcaRiaaigdaaiaawIcacaGLPaaa daahaaqabKqbGeaacaaIYaaaaaaaaaa@4F9B@

C.V= σ μ 1 = θ 4 +16 θ 2 +12 θ 2 +6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacqaH8oqBdaWg aaqcfasaaiaaigdaaKqbagqaamaaCaaabeqaaiadacUHYaIOaaaaai abg2da9maalaaabaWaaOaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI 0aaaaKqbakabgUcaRiaaigdacaaI2aGaeqiUde3aaWbaaeqajuaiba GaaGOmaaaajuaGcqGHRaWkcaaIXaGaaGOmaaqabaaabaGaeqiUde3a aWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI2aaaaaaa@537A@

C.V= σ μ 1 = θ 4 +4 θ 2 +2 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacqaH8oqBdaWg aaqcfasaaiaaigdaaKqbagqaamaaCaaabeqaaiadacUHYaIOaaaaai abg2da9maalaaabaWaaOaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI 0aaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabKqbGeaacaaIYa aaaKqbakabgUcaRiaaikdaaeqaaaqaaiabeI7aXnaaCaaabeqcfasa aiaaikdaaaqcfaOaey4kaSIaaGOmaaaaaaa@51FE@

β 1 = 2( θ 6 +30 θ 4 +36 θ 2 +24 ) ( θ 4 +16 θ 2 +12 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaeyypa0Za aSaaaeaacaaIYaWaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaI2a aaaKqbakabgUcaRiaaiodacaaIWaGaeqiUde3aaWbaaeqajuaibaGa aGinaaaajuaGcqGHRaWkcaaIZaGaaGOnaiabeI7aXnaaCaaabeqcfa saaiaaikdaaaqcfaOaey4kaSIaaGOmaiaaisdaaiaawIcacaGLPaaa aeaadaqadaqaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaey 4kaSIaaGymaiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqb akabgUcaRiaaigdacaaIYaaacaGLOaGaayzkaaWaaWbaaeqajuaiba GaaG4maiaac+cacaaIYaaaaaaaaaa@5D8F@

< β 1 = 2( θ 6 +6 θ 4 +6 θ 2 +2 ) ( θ 4 +4 θ 2 +2 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaeyypa0Za aSaaaeaacaaIYaWaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaI2a aaaKqbakabgUcaRiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaI0aaa aKqbakabgUcaRiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaK qbakabgUcaRiaaikdaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7a XnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaGinaiabeI7aXn aaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaaGaayjkaiaa wMcaamaaCaaabeqcfasaaiaaiodacaGGVaGaaGOmaaaaaaaaaa@59E5@

β 2 = 3( 3 θ 8 +128 θ 6 +408 θ 4 +576 θ 2 +240 ) ( θ 4 +16 θ 2 +12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaabaeqabaGaaG4maiabeI7aXnaaCaaabeqcfasaaiaaiIdaaa qcfaOaey4kaSIaaGymaiaaikdacaaI4aGaeqiUde3aaWbaaeqajuai baGaaGOnaaaajuaGcqGHRaWkcaaI0aGaaGimaiaaiIdacqaH4oqCda ahaaqabKqbGeaacaaI0aaaaaqcfayaaiabgUcaRiaaiwdacaaI3aGa aGOnaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaG OmaiaaisdacaaIWaaaaiaawIcacaGLPaaaaeaadaqadaqaaiabeI7a XnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaGymaiaaiAdacq aH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaaigdacaaI YaaacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGOmaaaaaaaaaa@6586@

β 2 = 3( 3 θ 8 +24 θ 6 +44 θ 4 +32 θ 2 +8 ) ( θ 4 +4 θ 2 +2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaqaaiaaiodacqaH4oqCdaahaaqabKqbGeaacaaI4aaaaKqbak abgUcaRiaaikdacaaI0aGaeqiUde3aaWbaaeqajuaibaGaaGOnaaaa juaGcqGHRaWkcaaI0aGaaGinaiabeI7aXnaaCaaabeqcfasaaiaais daaaqcfaOaey4kaSIaaG4maiaaikdacqaH4oqCdaahaaqabKqbGeaa caaIYaaaaKqbakabgUcaRiaaiIdaaiaawIcacaGLPaaaaeaadaqada qaaiabeI7aXnaaCaaabeqcfasaaiaaisdaaaqcfaOaey4kaSIaaGin aiabeI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaa GaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@6051@

γ= σ 2 μ 1 = θ 4 +16 θ 2 +12 θ( θ 2 +2 )( θ 2 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqaba Gamai4gkdiIcaaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqb GeaacaaI0aaaaKqbakabgUcaRiaaigdacaaI2aGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaIXaGaaGOmaaqaaiabeI7a XnaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHRa WkcaaIYaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCdaahaaqabKqb GeaacaaIYaaaaKqbakabgUcaRiaaiAdaaiaawIcacaGLPaaaaaaaaa@5E0C@

γ= σ 2 μ 1 = θ 4 +4 θ 2 +2 θ( θ 2 +1 )( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqaba Gamai4gkdiIcaaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqb GeaacaaI0aaaaKqbakabgUcaRiaaisdacqaH4oqCdaahaaqabKqbGe aacaaIYaaaaKqbakabgUcaRiaaikdaaeaacqaH4oqCdaqadaqaaiab eI7aXnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGymaaGaay jkaiaawMcaamaabmaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkcaaIYaaacaGLOaGaayzkaaaaaaaa@5C8F@

Table 1: Characteristics of Akash and Shanker Distributions.

Lindley Distribution

Exponential Distribution

f( t )= θ 2 θ+1 ( 1+t ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaGOmaaaaaKqbagaacqaH4oqCcqGHRaWkca aIXaaaamaabmaabaGaaGymaiabgUcaRiaadshaaiaawIcacaGLPaaa caWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlaadshaaa aaaa@4C2A@

f( t )=θ e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iabeI7aXjaadwga daahaaqabKqbGeaacqGHsislcqaH4oqCcaaMc8UaamiDaaaaaaa@430E@

F( t )=1 θ+1+θt θ+1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl daWcaaqaaiabeI7aXjabgUcaRiaaigdacqGHRaWkcqaH4oqCcaaMc8 UaamiDaaqaaiabeI7aXjabgUcaRiaaigdaaaGaamyzamaaCaaabeqc fasaaiabgkHiTiabeI7aXjaaykW7caWG0baaaaaa@4EB2@

F( t )=1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisl caWGLbWaaWbaaeqajuaibaGaeyOeI0IaeqiUdeNaaGPaVlaadshaaa aaaa@42E0@

h( t )= θ 2 ( 1+t ) θ+1+θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU de3aaWbaaeqajuaibaGaaGOmaaaajuaGdaqadaqaaiaaigdacqGHRa WkcaWG0baacaGLOaGaayzkaaaabaGaeqiUdeNaey4kaSIaaGymaiab gUcaRiabeI7aXjaaykW7caWG0baaaaaa@4AE7@

h( t )=θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamiAam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9iabeI7aXbaa@3CAF@

m( t )= θ+2+θt θ( θ+1+θt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiU deNaey4kaSIaaGOmaiabgUcaRiabeI7aXjaaykW7caWG0baabaGaeq iUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaIXaGaey4kaSIaeqiUdeNa aGPaVlaadshaaiaawIcacaGLPaaaaaaaaa@4F2C@

m( t )= 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyBam aabmaabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGym aaqaaiabeI7aXbaaaaa@3D7F@

μ 1 = θ+2 θ( θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaeqiUdeNaey4kaSIaaGOmaaqaaiabeI7aXn aabmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaaaaaaaa@47D8@

μ 1 = 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaadaahaaqabeaacWaGGBOmGika aiabg2da9maalaaabaGaaGymaaqaaiabeI7aXbaaaaa@4063@

μ 2 = θ 2 +4θ+2 θ 2 ( θ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiabeI7a XnaaCaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGinaiabeI7aXj abgUcaRiaaikdaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqb aoaabmaabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaaCa aabeqcfasaaiaaikdaaaaaaaaa@4C65@

μ 2 = 1 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiaaigda aeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaaaaaa@3E66@

C.V= σ μ 1 = θ 2 +4θ+2 θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacuaH8oqBgaqb amaaBaaajuaibaGaaGymaaqcfayabaaaaiabg2da9maalaaabaWaaO aaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakabgUcaRiaa isdacqaH4oqCcqGHRaWkcaaIYaaabeaaaeaacqaH4oqCcqGHRaWkca aIYaaaaaaa@4BCA@

C.V= σ μ 1 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qai aac6cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacuaH8oqBgaqb amaaBaaajuaibaGaaGymaaqcfayabaaaaiabg2da9iaaigdaaaa@40CD@

β 1 = 2( θ 3 +6 θ 2 +6θ+2 ) ( θ 2 +4θ+2 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaeyypa0Za aSaaaeaacaaIYaWaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIZa aaaKqbakabgUcaRiaaiAdacqaH4oqCdaahaaqabKqbGeaacaaIYaaa aKqbakabgUcaRiaaiAdacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaay zkaaaabaWaaeWaaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqb akabgUcaRiaaisdacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa WaaWbaaeqajuaibaqcfa4aaSGbaKqbGeaacaaIZaaabaGaaGOmaaaa aaaaaaaa@56C9@

β 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaGaeyypa0Ja aGOmaaaa@3B8F@

β 2 = 3( 3 θ 4 +24 θ 3 +44 θ 2 +32θ+8 ) ( θ 2 +4θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpdaWcaaqaaiaaioda daqadaabaeqabaGaaG4maiabeI7aXnaaCaaabeqcfasaaiaaisdaaa qcfaOaey4kaSIaaGOmaiaaisdacqaH4oqCdaahaaqabKqbGeaacaaI ZaaaaKqbakabgUcaRiaaisdacaaI0aGaeqiUde3aaWbaaeqajuaiba GaaGOmaaaaaKqbagaacqGHRaWkcaaIZaGaaGOmaiabeI7aXjabgUca RiaaiIdaaaGaayjkaiaawMcaaaqaamaabmaabaGaeqiUde3aaWbaae qajuaibaGaaGOmaaaajuaGcqGHRaWkcaaI0aGaeqiUdeNaey4kaSIa aGOmaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaikdaaaaaaaaa@5D19@

β 2 =9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaacqGH9aqpcaaI5aaaaa@3B87@

γ= σ 2 μ 1 = θ 2 +4θ+2 θ( θ+1 )( θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqaba Gamai4gkdiIcaaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaqabKqb GeaacaaIYaaaaKqbakabgUcaRiaaisdacqaH4oqCcqGHRaWkcaaIYa aabaGaeqiUde3aaeWaaeaacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGa ayzkaaWaaeWaaeaacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaa aaaaaa@57BF@

γ= σ 2 μ 1 = 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC Maeyypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc fayaaiabeY7aTnaaBaaajuaibaGaaGymaaqcfayabaWaaWbaaeqaba Gamai4gkdiIcaaaaGaeyypa0ZaaSaaaeaacaaIXaaabaGaeqiUdeha aaaa@467D@

Table 2: Characteristics of Lindley and Exponential Distributions.

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ for Akash Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaacaGGNaaaaa@3A7D@

299.990

59.950

29.900

9.713

5.556

2.333

1.294

0.833

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@

30001.000

1200.996

300.985

34.208

12.691

3.222

1.306

0.639

CV

0.577

0.578

0.580

0.602

0.641

0.769

0.883

0.959

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@

1.155

1.153

1.149

1.115

1.084

1.165

1.388

1.614

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@

5.000

4.997

4.987

4.897

4.785

4.834

5.473

6.391

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@

100.007

20.033

10.066

3.522

2.284

1.381

1.009

0.767

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  for Shanker Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaacaGGNaaaaa@3A7D@

199.990

39.950

19.901

6.391

3.600

1.500

0.872

0.600

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@

20000.000

799.998

199.990

22.146

7.840

1.750

0.676

0.340

CV

0.707

0.708

0.711

0.736

0.778

0.882

0.943

0.972

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@

1.414

1.414

1.414

1.421

1.452

1.620

1.779

1.876

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@

6.000

6.000

6.000

6.020

6.121

6.796

7.593

8.159

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@

100.005

20.025

10.049

3.465

2.178

1.167

0.775

0.567

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@  for Lindley Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaacaGGNaaaaa@3A7D@

199.010

39.048

19.091

5.897

3.333

1.500

0.933

0.667

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@

19999.020

799.093

199.174

21.631

7.556

1.750

0.729

0.389

CV

0.711

0.724

0.739

0.789

0.825

0.882

0.915

0.935

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@

1.414

1.417

1.422

1.464

1.512

1.620

1.699

1.756

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@

6.000

6.007

6.025

6.162

6.343

6.796

7.173

7.469

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@

100.493

20.465

10.433

3.668

2.267

1.167

0.781

0.583

Table 3: Values of μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIXaaajuaGbeaacaGGNaaaaa@3A7D@ , μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiVd0 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39D3@ , CV, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@ of Akash, Shanker and Lindley distributions for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Distribution

Over-Dispersion
( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH8aapcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E24@

Equi-Dispersion
( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH9aqpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E26@

Under-Dispersion
( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH8oqBcqGH+aGpcqaHdpWCdaahaaqabKqbGeaacaaIYaaaaaqc faOaayjkaiaawMcaaaaa@3E28@

Akash

θ<1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414D@

θ=1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@414F@

θ>1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaI1aGaaGymaiaaiwdacaaI0aGaaGim aiaaicdacaaIWaGaaGOnaiaaiodaaaa@4151@

Shanker

θ<1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415A@

θ=1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415C@

θ>1.171535555 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIXaGaaG4naiaaigdacaaI1aGaaG4m aiaaiwdacaaI1aGaaGynaiaaiwdaaaa@415E@

Lindley

θ<1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@415E@

θ=1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4160@

θ>1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGio aiaaiAdacaaI0aGaaGioaiaaiEdaaaa@4162@

Exponential

θ<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyipaWJaaGymaaaa@39F9@

θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde Naeyypa0JaaGymaaaa@39FB@

θ>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde NaeyOpa4JaaGymaaaa@39FD@

Table 4: Over-dispersion, equi-dispersion and under-dispersion of Akash, Shanker , Lindley and exponential distributions for varying values of their parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@ .

Figure 1: Graphs of C.V, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaae aacqaHYoGydaWgaaqcfasaaiaaigdaaKqbagqaaaqabaaaaa@39CD@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqOSdi 2aaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@39BE@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@  of Akash, Shanker and Lindley distributions for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiUde haaa@383A@ .

Model

Parameter Estimate

-2ln L

AIC

AICC

BIC

K-S Statistic

Data 1

Akash

1.355445

163.73

165.73

165.79

169.93

0.355

Shanker

0.956264

162.28

164.28

164.34

166.42

0.346

Lindley

0.996116

162.56

164.56

164.62

166.70

0.371

Exponential

0.663647

177.66

179.66

179.73

181.80

0.402

Data 2

Akash

0.043876

950.97

952.97

953.01

955.58

0.184

Shanker

0.029252

980.97

982.97

983.01

985.57

0.238

Lindley

0.028859

983.11

985.11

985.15

987.71

0.242

Exponential

0.014635

1044.87

1046.87

1046.91

1049.48

0.357

Data 3

Akash

0.041510

227.06

229.06

229.25

230.20

0.107

Shanker

0.027675

231.06

233.06

233.25

234.19

0.145

Lindley

0.027321

231.47

233.47

233.66

234.61

0.149

Exponential

0.013845

242.87

244.87

245.06

246.01

0.263

Data 4

Akash

0.013514

1255.83

1257.83

1257.87

1260.43

0.110

Shanker

0.009009

1251.19

1253.34

1253.38

1255.60

0.097

Lindley

0.008970

1251.34

1253.34

1253.38

1255.95

0.098

Exponential

0.004505

1280.52

1282.52

1282.56

1285.12

0.190

Data 5

Akash

0.030045

794.70

796.70

796.76

798.98

0.184

Shanker

0.020031

788.57

790.57

790.63

792.28

0.133

Lindley

0.019841

789.04

791.04

791.10

793.32

0.134

Exponential

0.010018

806.88

808.88

808.94

811.16

0.198

Data 6

Akash

0.119610

981.28

983.28

983.31

986.18

0.393

Shanker

0.079746

1033.10

1035.10

1035.13

1037.99

0.442

Lindley

0.077247

1041.64

1043.64

1043.68

1046.54

0.448

Exponential

0.040060

1130.26

1132.26

1132.29

1135.16

0.525

Data 7

Akash

0.013263

803.96

805.96

806.02

810.01

0.298

Shanker

0.008843

764.62

766.62

766.69

768.06

0.246

Lindley

0.008804

763.75

765.75

765.82

767.81

0.245

Exponential

0.004421

744.87

746.87

746.94

748.93

0.166

Data 8

Akash

0.013423

609.93

611.93

612.02

613.71

0.280

Shanker

0.008949

579.51

581.51

581.60

583.29

0.220

Lindley

0.008910

579.16

581.16

581.26

582.95

0.219

Exponential

0.004475

564.02

566.02

566.11

567.80

0.145

Data 9

Akash

0.310500

887.89

889.89

889.92

892.74

0.198

Shanker

0.210732

847.37

849.37

849.40

852.22

0.132

Lindley

0.196045

839.06

841.06

841.09

843.91

0.116

Exponential

0.106773

828.68

830.68

830.72

833.54

0.077

Data 10

Akash

0.050293

354.88

356.88

357.02

358.28

0.421

Shanker

0.033569

325.74

327.74

327.88

329.14

0.351

Lindley

0.033021

323.27

325.27

325.42

326.67

0.345

Exponential

0.016779

305.26

307.26

307.40

308.66

0.213

Data 11

Akash

1.165719

115.15

117.15

117.28

118.68

0.156

Shanker

0.853374

112.91

114.91

115.03

116.44

0.131

Lindley

0.823821

112.61

114.61

114.73

116.13

0.133

Exponential

0.532081

110.91

112.91

113.03

114.43

0.089

Data 12

Akash

0.295277

641.93

643.93

643.95

646.51

0.100

Shanker

0.198317

635.26

637.26

637.30

639.86

0.042

Lindley

0.186571

638.07

640.07

640.12

642.68

0.058

Exponential

0.101245

658.04

660.04

660.08

662.65

0.163

Data 13

Akash

0.024734

194.30

196.30

196.61

197.01

0.456

Shanker

0.016492

181.58

183.58

183.89

184.29

0.388

Lindley

0.016360

181.34

183.34

183.65

184.05

0.386

Exponential

0.008246

173.94

175.94

176.25

176.65

0.277

Data 14

Akash

1.156923

59.52

61.52

61.74

62.51

0.320

Shanker

0.803867

59.78

61.78

61.22

62.77

0.325

Lindley

0.816118

60.50

62.50

62.72

63.49

0.341

Exponential

0.526316

65.67

67.67

67.90

68.67

0.389

Data 15

Akash

0.097062

240.68

242.68

242.82

244.11

0.266

Shanker

0.064712

252.35

254.35

254.49

255.78

0.326

Lindley

0.062988

253.99

255.99

256.13

257.42

0.333

Exponential

0.032455

274.53

276.53

276.67

277.96

0.426

Data 16

Akash

0.964726

224.28

226.28

226.34

228.51

0.348

Shanker

0.658029

233.01

235.01

235.06

237.24

0.355

Lindley

0.659000

238.38

240.38

240.44

242.61

0.390

Exponential

0.407941

261.74

263.74

263.80

265.97

0.434

Table 5: MLE’s, -2ln L, AIC, AICC, BIC, K-S Statistics of the fitted distributions of data-sets 1-16.

0.55

0.93

1.25

1.36

1.49

1.52

1.58

1.61

1.64

1.68

1.73

1.81

2.00

0.74

1.04

1.27

1.39

1.49

1.53

1.59

1.61

1.66

1.68

1.76

1.82

2.01

0.77

1.11

1.28

1.42

1.50

1.54

1.60

1.62

1.66

1.69

1.76

1.84

2.24

0.81

1.13

1.29

1.48

1.50

1.55

1.61

1.62

1.66

1.70

1.77

1.84

0.84

1.24

1.30

1.48

1.51

1.55

1.61

1.63

1.67

1.70

1.78

1.89

Data Set 1: The data set represents the strength of 1.5cm glass fibers measured at the National Physical Laboratory, England. Unfortunately, the units of measurements are not given in the paper, and they are taken from Smith & Naylor [19].

5

25

31

32

34

35

38

39

39

40

42

43

43

43

44

44

47

48

48

49

49

49

51

54

55

55

55

56

56

56

58

59

59

59

59

59

63

63

64

64

65

65

65

66

66

66

66

66

67

67

67

68

69

69

69

69

71

71

72

73

73

73

74

74

76

76

77

77

77

77

77

77

79

79

80

81

83

83

84

86

86

87

90

91

92

92

92

92

93

94

97

98

98

99

101

103

105

109

136

147

Data Set 2: The data is given by Birnbaum & Saunders [20] on the fatigue life of 6061 – T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data (x10-3 ) are presented below (after subtracting 65).

17.88

28.92

33.00

41.52

42.12

45.60

48.80

51.84

51.96

54.12

55.56

67.80

68.44

68.64

68.88

84.12

93.12

98.64

105.12

105.84

127.92

128.04

173.40

Data Set 3: The data set is from Lawless (1982, p-228). The data given arose in tests on endurance of deep groove ball bearings. The data are the number of million revolutions before failure for each of the 23 ball bearings in the life tests.

86

146

251

653

98

249

400

292

131

169

175

176

76

264

15

364

195

262

88

264

157

220

42

321

180

198

38

20

61

121

282

224

149

180

325

250

196

90

229

166

38

337

65

151

341

40

40

135

597

246

211

180

93

315

353

571

124

279

81

186

497

182

423

185

229

400

338

290

398

71

246

185

188

568

55

55

61

244

20

284

393

396

203

829

239

236

286

194

277

143

198

264

105

203

124

137

135

350

193

188

Data Set 4: The data is from Picciotto [21] and arose in test on the cycle at which the Yarn failed. The data are the number of cycles until failure of the yarn.

12

15

22

24

24

32

32

33

34

38

38

43

44

48

52

53

54

54

55

56

57

58

58

59

60

60

60

60

61

62

63

65

65

67

68

70

70

72

73

75

76

76

81

83

84

85

87

91

95

96

98

99

109

110

121

127

129

131

143

146

146

175

175

211

233

258

258

263

297

341

341

376

Data Set 5: This data represents the survival times (in days) of 72 guinna pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal [22].

19(16)

20(15)

21(14)

22(9)

23(12)

24(10)

25(6)

26(9)

27(8)

28(5)

29(6)

30(4)

31(3)

32(4)

33

34

35(4)

36(2)

37(2)

39

42

44

Data Set 6: This data is related with behavioral sciences, collected by Balakrishnan N et al. [23]: The scale “General Rating of Affective Symptoms for Preschoolers (GRASP)” measures behavioral and emotional problems of children, which can be classified with depressive condition or not according to this scale. A study conducted by the authors in a city located at the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis.

6.53

7

10.42

14.48

16.10

22.70

34

41.55

42

45.28

49.40

53.62

63

64

83

84

91

108

112

129

133

133

139

140

140

146

149

154

157

160

160

165

146

149

154

157

160

160

165

173

176

218

225

241

248

273

277

297

405

417

420

440

523

583

594

1101

1146

1417

Data Set 7: The data set reported by Efron [24] represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using radiotherapy (RT).

12.20

23.56

23.7

25.9

31.98

37

41.35

47.38

55.46

58.36

63.47

68.46

78.3

74.5

81.43

84

92

94

110

112

119

127

130

133

140

146

155

159

173

179

194

195

209

249

281

319

339

432

469

519

633

725

817

1776

Data Set 8: The data set reported by Efron [24] represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy (RT+CT).

0.08

2.09

3.48

4.87

6.94

8.66

13.11

23.63

0.20

2.23

3.52

4.98

6.97

9.02

13.29

0.40

2.26

3.57

5.06

7.09

9.22

13.80

25.74

0.50

2.46

3.64

5.09

7.26

9.47

14.24

25.82

0.51

2.54

3.70

5.17

7.28

9.74

14.76

6.31

0.81

2.62

3.82

5.32

7.32

10.06

14.77

32.15

2.64

3.88

5.32

7.39

10.34

14.83

34.26

0.90

2.69

4.18

5.34

7.59

10.66

15.96

36.66

1.05

2.69

4.23

5.41

7.62

10.75

16.62

43.01

1.19

2.75

4.26

5.41

7.63

17.12

46.12

1.26

2.83

4.33

5.49

7.66

11.25

17.14

79.05

1.35

2.87

5.62

7.87

11.64

17.36

1.40

3.02

4.34

5.71

7.93

11.79

18.10

1.46

4.40

5.85

8.26

11.98

19.13

1.76

3.25

4.50

6.25

8.37

12.02

2.02

3.31

4.51

6.54

8.53

12.03

20.28

2.02

3.36

6.76

12.07

21.73

2.07

3.36

6.93

8.65

12.63

22.69

Data set 9: This data set represents remission times (in months) of a random sample of 128 bladder cancer patients reported in Lee & Wang [25].

23

261

87

7

120

14

62

47

225

71

246

21

42

20

5

12

120

11

3

14

71

11

14

11

16

90

1

16

52

95

Data Set 10: This data set is given by Linhart & Zucchini [26], which represents the failure times of the air conditioning system of an airplane.

5.1

1.2

1.3

0.6

0.5

2.4

0.5

1.1

8

0.8

0.4

0.6

0.9

0.4

2

0.5

5.3

3.2

2.7

2.9

2.5

2.3

1

0.2

0.1

0.1

1.8

0.9

2

4

6.8

1.2

0.4

0.2

Data Set 11: This data set used by Bhaumik et al. [27], is vinyl chloride data obtained from clean up gradient monitoring wells in mg/l.

0.8

0.8

1.3

1.5

1.8

1.9

1.9

2.1

2.6

2.7

2.9

3.1

3.2

3.3

3.5

3.6

4.0

4.1

4.2

4.2

4.3

4.3

4.4

4.4

4.6

4.7

4.7

4.8

4.9

4.9

5.0

5.3

5.5

5.7

5.7

6.1

6.2

6.2

6.2

6.3

6.7

6.9

7.1

7.1

7.1

7.1

7.4

7.6

7.7

8.0

8.2

8.6

8.6

8.6

8.8

8.8

8.9

8.9

9.5

9.6

9.7

9.8

10.7

10.9

11.0

11.0

11.1

11.2

11.2

11.5

11.9

12.4

12.5

12.9

13.0

13.1

13.3

13.6

13.7

13.9

14.1

15.4

15.4

17.3

17.3

18.1

18.2

18.4

18.9

19.0

19.9

20.6

21.3

21.4

21.9

23.0

27.0

31.6

33.1

38.5

Data set 12: This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al. [8] for fitting the Lindley [7] distribution.

74

57

48

29

502

12

70

21

29

386

59

27

153

26

326

Data Set 13: This data is for the times between successive failures of air conditioning equipment in a Boeing 720 airplane, Proschan [28].

1.1

1.4

1.3

1.7

1.9

1.8

1.6

2.2

1.7

2.7

4.1

1.8

1.5

1.2

1.4

3

1.7

2.3

1.6

2

Data set 14: This data set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross & Clark [29].

18.83

20.8

21.657

23.03

23.23

24.05

24.321

25.5

25.52

25.8

26.69

26.77

26.78

27.05

27.67

29.9

31.11

33.2

33.73

33.76

33.89

34.76

35.75

35.91

36.98

37.08

37.09

39.58

44.045

45.29

45.381

Data Set 15: This data set is the strength data of glass of the aircraft window reported by Fuller et al. [30].

1.312

1.314

1.479

1.552

1.700

1.803

1.861

1.865

1.944

1.958

1.966

1.997

2.006

2.021

2.027

2.055

2.063

2.098

2.140

2.179

2.224

2.240

2.253

2.270

2.272

2.274

2.301

2.301

2.359

2.382

2.382

2.426

2.434

2.435

2.478

2.490

2.511

2.514

2.535

2.554

2.566

2.570

2.586

2.629

2.633

2.642

2.648

2.684

2.697

2.726

2.770

2.773

2.800

2.809

2.818

2.821

2.848

2.880

2.954

3.012

3.067

3.084

3.090

3.096

3.128

3.233

3.433

3.585

3.858

Data Set 16: The following data represent the tensile strength, measured in GPa, of 69 carbon fibers tested under tension at gauge lengths of 20mm Bader & Priest [31,32].

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