ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 4 Issue 1 - 2016
On Modeling of Lifetime Data Using Aradhana, Sujatha, 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 25, 2016 | Published: July 07, 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 Aradhana, Sujatha, Lindley and Exponential Distributions. Biom Biostat Int J 4(1): 00087. DOI: 10.15406/bbij.2016.04.00087

Abstract

The modeling and statistical analysis of lifetime data are crucial for statisticians and research workers in almost all applied sciences including engineering, medical sciences/biological sciences, insurance, finance, amongst others. One parameter lifetime distributions that are popular in Statistics literature for modeling lifetime data are exponential and Lindley distributions. An extensive study has been carried out by Shanker et al. [1] for modeling lifetime data using Lindley and exponential distributions and 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 one parameter Lifetime distributions namely “Aradhana distribution” and “Sujatha distribution” for modeling lifetime data.

In the present paper the interrelationships and comparative studies of Aradhana, Sujatha, Lindley and exponential distributions have been made to model lifetime data. The relationships, their distributional properties and estimation of parameter have been discussed. The applications and goodness of fit of these distributions for modeling lifetime data through various examples from engineering, medical science and other fields have also been discussed and explained.

Keywords: Aradhana distribution; Sujatha distribution; Lindley distribution; Exponential distribution; Statistical properties; Estimation of parameter; Goodness of fit

Introduction

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

Shanker [2,3] has introduced one parameter continuous distributions named, “Aradhana distribution” and “Sujatha distribution”for modeling lifetime data from engineering and medical science and studied its various mathematical properties, estimation of its parameter, and its applications. A number of continuous distributions for modeling lifetime data have been introduced in statistical literature including exponential, Lindley, gamma, lognormal and Weibull, amongst others. The exponential, Lindley and the Weibull distributions are more popular in practice than the gamma and the lognormal distributions because the survival functions of the gamma and the lognormal distributions cannot be expressed in closed forms and both require numerical integration. Though Aradhana, Sujatha, Lindley and exponential distributions are of one parameter, Aradhana, Sujatha and Lindley distributions have advantage over the exponential distribution that the exponential distribution has constant hazard rate and mean residual life function whereas the Aradhana, Sujatha, and Lindley distributions have increasing hazard rate and decreasing mean residual life function. Further, Aradhana and Sujatha distributions of Shanker [2,3] have flexibility over both Lindley and exponential distributions.

Aradhana, Sujatha, Lindley and Exponential Distributions

Shanker [2] introduced a new one parameter continuous distribution named, ‘Aradhana distribution’ for modeling lifetime data from engineering and medical science. This distribution is a three- component mixture of an exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@  distribution, a gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@ 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 +2θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaaGOmaiabeI7aXjabgU caRiaaikdaaaaaaa@4226@  , 2θ θ 2 +2θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaGaeqiUdehabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaa juaGcqGHRaWkcaaIYaGaeqiUdeNaey4kaSIaaGOmaaaaaaa@4148@  and 2 θ 2 +2θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcaaIYaGaeqiUdeNaey4kaSIaaGOmaaaaaaa@3F92@  respectively. It has been shown by Shanker [2] that Aradhana distribution is flexible than the Lindley distribution for modeling lifetime data in reliability and in terms of its hazard rate shapes and it gives better fit than Akash, Shanker, Lindley and exponential distributions in modeling lifetime data. 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 [4] has also obtained a Poisson mixture of Aradhana distribution named, “Poisson-Aradhana distribution (PAD)”for modeling count data.

Shanker [3] introduced another one parameter continuous distribution named, ‘Sujatha distribution’ for modeling lifetime data from engineering and medical science. This distribution is also a three-component mixture of an exponential ( θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacqaH4oqCaiaawIcacaGLPaaaaaa@39C3@  distribution, a gamma ( 2,θ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaaIYaGaaiilaiabeI7aXbGaayjkaiaawMcaaaaa@3B2F@ 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 +θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaaqcfayaaiabeI7aXnaa CaaabeqcfasaaiaaikdaaaqcfaOaey4kaSIaeqiUdeNaey4kaSIaaG Omaaaaaaa@416A@ , θ θ 2 +θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaH4oqCaeaacqaH4oqCdaahaaqabKqbGeaacaaIYaaaaKqbakab gUcaRiabeI7aXjabgUcaRiaaikdaaaaaaa@3FD0@ and 2 θ 2 +θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacaaIYaaabaGaeqiUde3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGH RaWkcqaH4oqCcqGHRaWkcaaIYaaaaaaa@3ED6@ respectively. It has been shown by Shanker [3] that Sujatha distribution is flexible than the Lindley distribution for modeling lifetime data in reliability and in terms of its hazard rate shapes and it gives better fit than Lindley and exponential distributions in modeling lifetime data. 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 [5] has also obtained a Poisson mixture of Sujatha distribution named, “Poisson-Sujatha distribution (PSD)”for modeling count data.

The Lindley 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 and is given by Lindley [6] in the context of Bayesian Statistics as a counter example of fiducial Statistics. 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. [7]. The Lindley distribution has been generalized, extended, mixed, modified and its detailed applications in reliability and other fields of knowledge by different researchers including Sankaran [8], Zakerzadeh & Dolati [9], Nadarajah et al. [10], Deniz & Ojeda [11], Bakouch et al. [12], Shanker & Mishra [13,14,15], Shanker & Amanuel [16], Ghitany et al. [17], Shanker et al. [1,18-21], are some among others.

In statistical literature, exponential distribution was the first widely used lifetime distribution model in areas ranging from studies on the lifetimes of manufactured items 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 suitable for representing the lifetimes of many phenomenons such as various types of manufactured items.

Let T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaaaa@36CF@ be a continuous random variable representing the lifetimes of individuals in some population. 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 Aradhana and Sujatha distributions are summarized in Table 1 and of Lindley and exponential distributions 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada Gcaaqaaiabek7aInaaBaaaleaacaaIXaaabeaaaeqaaaGccaGLOaGa ayzkaaaaaa@3A21@ , coefficient of Kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHYoGydaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaaa@3A12@ , and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHZoWzaiaawIcacaGLPaaaaaa@3926@  for Aradhana, Sujatha and Lindley distributions for various values of their parameter for comparative study are summarized in the Table 3.

The condition under which Aradhana, Sujatha, Lindley and exponential distributions are Over-dispersion ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH8aapcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CEF@ , equi-dispersion ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH9aqpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF1@  and under-dispersion ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH+aGpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF3@  of Aradhana, Sujatha, 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@ are presented in Table 4.

Graphs of coefficient of variation (C.V), coefficient of skewness ( β 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaaaaa@388F@ ) coefficient of kurtosis ( β 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaaaa@3880@ ) and index of dispersion ( γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379E@ ) for Aradhana, Sujatha, and Lindley distributions are presented for varying values of their parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  in Figure 1.

Estimation of Parameter

    1. Estimate of the parameter of Aradhana distribution

Let ( t 1 , t 2 , t 3 ,..., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaaykW7caWG0bWaaSba aSqaaiaaikdaaeqaaOGaaiilaiaaykW7caWG0bWaaSbaaSqaaiaaio daaeqaaOGaaiilaiaaykW7caaMc8UaaiOlaiaac6cacaGGUaGaaGPa VlaaykW7caGGSaGaamiDamaaBaaaleaacaWGUbaabeaaaOGaayjkai aawMcaaaaa@4D7A@  be a random sample from Aradhana distribution. The maximum likelihood estimate (MLE) θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  and the method of moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aaaaa@37BB@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  is the solution of the following cubic equation

t ¯ θ 3 +( 2 t ¯ 1 ) θ 2 +2( t ¯ 2 )θ6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaara GaeqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaSYaaeWaaeaacaaI YaGabmiDayaaraGaeyOeI0IaaGymaaGaayjkaiaawMcaaiabeI7aXn aaCaaaleqabaGaaGOmaaaakiabgUcaRiaaikdadaqadaqaaiqadsha gaqeaiabgkHiTiaaikdaaiaawIcacaGLPaaacqaH4oqCcqGHsislca aI2aGaeyypa0JaaGimaaaa@4D3E@ .

Estimate of the parameter of Sujatha distribution

Let ( t 1 , t 2 , t 3 ,..., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaaykW7caWG0bWaaSba aSqaaiaaikdaaeqaaOGaaiilaiaaykW7caWG0bWaaSbaaSqaaiaaio daaeqaaOGaaiilaiaaykW7caaMc8UaaiOlaiaac6cacaGGUaGaaGPa VlaaykW7caGGSaGaamiDamaaBaaaleaacaWGUbaabeaaaOGaayjkai aawMcaaaaa@4D7A@  be random sample from Sujatha distribution. The maximum likelihood estimate (MLE) θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  and the method of moment estimate (MOME) θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aaaaa@37BB@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  is the solution of the following cubic equation

  t ¯ θ 3 +( t ¯ 1 ) θ 2 +2( t ¯ 1 )θ6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaara GaeqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaSYaaeWaaeaaceWG 0bGbaebacqGHsislcaaIXaaacaGLOaGaayzkaaGaeqiUde3aaWbaaS qabeaacaaIYaaaaOGaey4kaSIaaGOmamaabmaabaGabmiDayaaraGa eyOeI0IaaGymaaGaayjkaiaawMcaaiabeI7aXjabgkHiTiaaiAdacq GH9aqpcaaIWaaaaa@4C81@

Estimate of the parameter of Lindley distribution

Let ( t 1 , t 2 ,...., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadshadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcaca WG0bWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaaa@424E@  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 MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  and MOME θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aaaaa@37BB@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  is given by θ ^ = ( t ¯ 1 )+ ( t ¯ 1 ) 2 +8 t ¯ 2 t ¯ ; t ¯ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpdaWcaaqaaiabgkHiTmaabmaabaGabmiDayaaraGaeyOe I0IaaGymaaGaayjkaiaawMcaaiabgUcaRmaakaaabaWaaeWaaeaace WG0bGbaebacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaa caaIYaaaaOGaey4kaSIaaGioaiaaykW7ceWG0bGbaebaaSqabaaake aacaaIYaGaaGPaVlqadshagaqeaaaacaGG7aGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UabmiDayaaraGaeyOpa4JaaGimaaaa@58A9@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaara aaaa@3707@ is the sample mean.

Estimate of the parameter of Exponential distribution

Let ( t 1 , t 2 ,...., t n ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG0bWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadshadaWgaaWcbaGa aGOmaaqabaGccaGGSaGaaiOlaiaac6cacaGGUaGaaiOlaiaacYcaca WG0bWaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaaaaa@424E@  be a random sample of size n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@  from exponential distribution. The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aaaaa@37BC@  of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  and MOME θ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaG aaaaa@37BB@ of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@  is given by θ ^ = 1 t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqiUdeNbaK aacqGH9aqpdaWcaaqaaiaaigdaaeaaceWG0bGbaebaaaaaaa@3A9E@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaara aaaa@3707@ is the sample mean.

Applications and Goodness of fit

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

In order to compare Aradhana, Sujatha, Lindley and exponential distributions, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiGacYgacaGGUbGaamitaaaa@3A54@ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics ( Kolmogorov-Smirnov Statistics) for all 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadM eacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamitaiab gUcaRiaaikdacaWGRbaaaa@4044@ , AICC=AIC+ 2k( k+1 ) ( nk1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaadM eacaWGdbGaam4qaiabg2da9iaadgeacaWGjbGaam4qaiabgUcaRmaa laaabaGaaGOmaiaadUgadaqadaqaaiaadUgacqGHRaWkcaaIXaaaca GLOaGaayzkaaaabaWaaeWaaeaacaWGUbGaeyOeI0Iaam4AaiabgkHi TiaaigdaaiaawIcacaGLPaaaaaaaaa@4931@ , BIC=2lnL+klnn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiaadM eacaWGdbGaeyypa0JaeyOeI0IaaGOmaiGacYgacaGGUbGaamitaiab gUcaRiaadUgaciGGSbGaaiOBaiaad6gaaaa@4260@  and

D= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiraiabg2 da9maaxababaGaae4uaiaabwhacaqGWbaaleaacaWG4baabeaakmaa emaabaGaamOramaaBaaaleaacaWGUbaabeaakmaabmaabaGaamiEaa GaayjkaiaawMcaaiabgkHiTiaadAeadaWgaaWcbaGaaGimaaqabaGc daqadaqaaiaadIhaaiaawIcacaGLPaaaaiaawEa7caGLiWoaaaa@4890@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@36E6@ = number of parameters, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@ = the sample size and F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaaBa aaleaacaWGUbaabeaakmaabmaabaGaamiEaaGaayjkaiaawMcaaaaa @3A70@ is the empirical distribution function. The best distribution is the distribution corresponding to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0IaaG OmaiGacYgacaGGUbGaamitaaaa@3A54@ , 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

Concluding Remarks

In this paper an attempt has been made to find the suitability of Aradhana, Sujatha, Lindley and exponential distributions for modeling real lifetime data from engineering, medical science and other fields. Firstly a table for values of the various characteristics of Aradhana, Sujatha, Lindley and exponential distributions has been presented for different values of their parameter which reflects their nature and behavior. The condition under which Aradhana, Sujatha, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed has been given. Several lifetime data from medical science, engineering and other fields of knowledge have been fitted using Aradhana, Sujatha, Lindley and exponential distributions to study the advantages and disadvantages of these distributions. The goodness of fit test of these distributions using Kolmogorov-Smirnov tests indicate that each has advantages and disadvantages for modeling lifetime data.

Aradhana Distribution

Sujatha Distribution

f( t )= θ 3 θ 2 +2θ+2 ( 1+t ) 2 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIZaaaaaGcbaGaeqiUde3aaWbaaSqabeaacaaIYa aaaOGaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaikdaaaWaaeWaaeaa caaIXaGaey4kaSIaamiDaaGaayjkaiaawMcaamaaCaaaleqabaGaaG OmaaaakiaadwgadaahaaWcbeqaaiabgkHiTiabeI7aXjaaykW7caWG 0baaaaaa@500E@

f( t )= θ 3 θ 2 +θ+2 ( 1+t+ t 2 ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIZaaaaaGcbaGaeqiUde3aaWbaaSqabeaacaaIYa aaaOGaey4kaSIaeqiUdeNaey4kaSIaaGOmaaaadaqadaqaaiaaigda cqGHRaWkcaWG0bGaey4kaSIaamiDamaaCaaaleqabaGaaGOmaaaaaO GaayjkaiaawMcaaiaadwgadaahaaWcbeqaaiabgkHiTiabeI7aXjaa ykW7caWG0baaaaaa@512D@

F( t )=1[ 1+ θt( θt+2θ+2 ) θ 2 +2θ+2 ] e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWa daqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaaykW7caWG0bWaae WaaeaacqaH4oqCcaaMc8UaamiDaiabgUcaRiaaikdacqaH4oqCcqGH RaWkcaaIYaaacaGLOaGaayzkaaaabaGaeqiUde3aaWbaaSqabeaaca aIYaaaaOGaey4kaSIaaGOmaiabeI7aXjabgUcaRiaaikdaaaaacaGL BbGaayzxaaGaamyzamaaCaaaleqabaGaeyOeI0IaeqiUdeNaaGPaVl aadshaaaGccaaMc8oaaa@5DED@

F( t )=1[ 1+ θt( θt+θ+2 ) θ 2 +θ+2 ] e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWa daqaaiaaigdacqGHRaWkdaWcaaqaaiabeI7aXjaaykW7caWG0bWaae WaaeaacqaH4oqCcaaMc8UaamiDaiabgUcaRiabeI7aXjabgUcaRiaa ikdaaiaawIcacaGLPaaaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaa GccqGHRaWkcqaH4oqCcqGHRaWkcaaIYaaaaaGaay5waiaaw2faaiaa dwgadaahaaWcbeqaaiabgkHiTiabeI7aXjaaykW7caWG0baaaOGaaG PaVdaa@5C75@

h( t )= θ 3 ( 1+t ) 2 θt( θt+2θ+2 )+( θ 2 +2θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIZaaaaOWaaeWaaeaacaaIXaGaey4kaSIaamiDaa GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaaiabeI7aXjaa ykW7caWG0bWaaeWaaeaacqaH4oqCcaaMc8UaamiDaiabgUcaRiaaik dacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaGaey4kaSYaaeWa aeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaeq iUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaaaaaaa@5B2C@

h( t )= θ 3 ( 1+t+ t 2 ) θ 2 ( 1+t+ t 2 )+2θt+θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIZaaaaOWaaeWaaeaacaaIXaGaey4kaSIaamiDai abgUcaRiaadshadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaa aeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaaigdacq GHRaWkcaWG0bGaey4kaSIaamiDamaaCaaaleqabaGaaGOmaaaaaOGa ayjkaiaawMcaaiabgUcaRiaaikdacqaH4oqCcaWG0bGaey4kaSIaeq iUdeNaey4kaSIaaGOmaaaaaaa@562B@

m( t )= θ 2 t 2 +2θt( θ+2 )+( θ 2 +4θ+6 ) θ[ θt( θt+2θ+2 )+( θ 2 +2θ+2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIYaaaaOGaaGPaVlaadshadaahaaWcbeqaaiaaik daaaGccqGHRaWkcaaIYaGaeqiUdeNaaGPaVlaadshadaqadaqaaiab eI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaacqGHRaWkdaqadaqaai abeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaisdacqaH4oqC cqGHRaWkcaaI2aaacaGLOaGaayzkaaaabaGaeqiUde3aamWaaeaacq aH4oqCcaaMc8UaamiDamaabmaabaGaeqiUdeNaaGPaVlaadshacqGH RaWkcaaIYaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaiabgU caRmaabmaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIa aGOmaiabeI7aXjabgUcaRiaaikdaaiaawIcacaGLPaaaaiaawUfaca GLDbaaaaaaaa@71FE@

m( t )= θ 2 ( t 2 +t+1 )+2θ( t+1 )+6 θ[ ( θ 2 +θ+2 )+θt( θt+θ+2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaWG0bWaaWbaaSqabeaaca aIYaaaaOGaey4kaSIaamiDaiabgUcaRiaaigdaaiaawIcacaGLPaaa cqGHRaWkcaaIYaGaeqiUde3aaeWaaeaacaWG0bGaey4kaSIaaGymaa GaayjkaiaawMcaaiabgUcaRiaaiAdaaeaacqaH4oqCdaWadaqaamaa bmaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqiUde Naey4kaSIaaGOmaaGaayjkaiaawMcaaiabgUcaRiabeI7aXjaaykW7 caWG0bWaaeWaaeaacqaH4oqCcaaMc8UaamiDaiabgUcaRiabeI7aXj abgUcaRiaaikdaaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaaaaa@6850@

μ 1 = θ 2 +4θ+6 θ( θ 2 +2θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOWaaWbaaSqabeaakiadacUHYaIOaaGaeyyp a0ZaaSaaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkca aI0aGaeqiUdeNaey4kaSIaaGOnaaqaaiabeI7aXnaabmaabaGaeqiU de3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGOmaiabeI7aXjabgU caRiaaikdaaiaawIcacaGLPaaaaaaaaa@4F4D@

μ 1 = θ 2 +2θ+6 θ( θ 2 +θ+2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOWaaWbaaSqabeaakiadacUHYaIOaaGaeyyp a0ZaaSaaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkca aIYaGaeqiUdeNaey4kaSIaaGOnaaqaaiabeI7aXnaabmaabaGaeqiU de3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqiUdeNaey4kaSIaaG OmaaGaayjkaiaawMcaaaaaaaa@4E8F@

μ 2 = θ 4 +8 θ 3 +24 θ 2 +24θ+12 θ 2 ( θ 2 +2θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWc beqaaiaaisdaaaGccqGHRaWkcaaI4aGaeqiUde3aaWbaaSqabeaaca aIZaaaaOGaey4kaSIaaGOmaiaaisdacqaH4oqCdaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIYaGaaGinaiabeI7aXjabgUcaRiaaigdaca aIYaaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacqaH 4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaeqiUdeNaey 4kaSIaaGOmaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaa aa@58D3@

μ 2 = θ 4 +4 θ 3 +18 θ 2 +12θ+12 θ 2 ( θ 2 +θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWc beqaaiaaisdaaaGccqGHRaWkcaaI0aGaeqiUde3aaWbaaSqabeaaca aIZaaaaOGaey4kaSIaaGymaiaaiIdacqaH4oqCdaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcaaIXaGaaGOmaiabeI7aXjabgUcaRiaaigdaca aIYaaabaGaeqiUde3aaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacqaH 4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcqaH4oqCcqGHRaWkca aIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaaa@5813@

C.V= σ μ 1 = θ 4 +8 θ 3 +24 θ 2 +24θ+12 θ 2 +4θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaac6 cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacqaH8oqBdaWgaaWc baGaaGymaaqabaGcdaahaaWcbeqaaOGamai4gkdiIcaaaaGaeyypa0 ZaaSaaaeaadaGcaaqaaiabeI7aXnaaCaaaleqabaGaaGinaaaakiab gUcaRiaaiIdacqaH4oqCdaahaaWcbeqaaiaaiodaaaGccqGHRaWkca aIYaGaaGinaiabeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa ikdacaaI0aGaeqiUdeNaey4kaSIaaGymaiaaikdaaSqabaaakeaacq aH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI0aGaeqiUdeNa ey4kaSIaaGOnaaaaaaa@5C2F@

C.V= σ μ 1 = θ 4 +4 θ 3 +18 θ 2 +12θ+12 θ 2 +2θ+6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaac6 cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacqaH8oqBdaWgaaWc baGaaGymaaqabaGcdaahaaWcbeqaaOGamai4gkdiIcaaaaGaeyypa0 ZaaSaaaeaadaGcaaqaaiabeI7aXnaaCaaaleqabaGaaGinaaaakiab gUcaRiaaisdacqaH4oqCdaahaaWcbeqaaiaaiodaaaGccqGHRaWkca aIXaGaaGioaiabeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa igdacaaIYaGaeqiUdeNaey4kaSIaaGymaiaaikdaaSqabaaakeaacq aH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaeqiUdeNa ey4kaSIaaGOnaaaaaaa@5C29@

β 1 = 2( θ 6 +12 θ 5 +54 θ 4 +100 θ 3 +108 θ 2 +72θ+24 ) ( θ 4 +8 θ 3 +24 θ 2 +24θ+12 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaakiabg2da9maalaaabaGa aGOmamaabmaaeaqabeaacqaH4oqCdaahaaWcbeqaaiaaiAdaaaGccq GHRaWkcaaIXaGaaGOmaiabeI7aXnaaCaaaleqabaGaaGynaaaakiab gUcaRiaaiwdacaaI0aGaeqiUde3aaWbaaSqabeaacaaI0aaaaOGaey 4kaSIaaGymaiaaicdacaaIWaGaeqiUde3aaWbaaSqabeaacaaIZaaa aaGcbaGaey4kaSIaaGymaiaaicdacaaI4aGaeqiUde3aaWbaaSqabe aacaaIYaaaaOGaey4kaSIaaG4naiaaikdacqaH4oqCcqGHRaWkcaaI YaGaaGinaaaacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4oqCdaahaa WcbeqaaiaaisdaaaGccqGHRaWkcaaI4aGaeqiUde3aaWbaaSqabeaa caaIZaaaaOGaey4kaSIaaGOmaiaaisdacqaH4oqCdaahaaWcbeqaai aaikdaaaGccqGHRaWkcaaIYaGaaGinaiabeI7aXjabgUcaRiaaigda caaIYaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaGaai4laiaaik daaaaaaaaa@70EF@

β 1 = 2( θ 6 +6 θ 5 +36 θ 4 +44 θ 3 +54 θ 2 +36θ+24 ) ( θ 4 +4 θ 3 +18 θ 2 +12θ+12 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaakiabg2da9maalaaabaGa aGOmamaabmaaeaqabeaacqaH4oqCdaahaaWcbeqaaiaaiAdaaaGccq GHRaWkcaaI2aGaeqiUde3aaWbaaSqabeaacaaI1aaaaOGaey4kaSIa aG4maiaaiAdacqaH4oqCdaahaaWcbeqaaiaaisdaaaGccqGHRaWkca aI0aGaaGinaiabeI7aXnaaCaaaleqabaGaaG4maaaaaOqaaiabgUca RiaaiwdacaaI0aGaeqiUde3aaWbaaSqabeaacaaIYaaaaOGaey4kaS IaaG4maiaaiAdacqaH4oqCcqGHRaWkcaaIYaGaaGinaaaacaGLOaGa ayzkaaaabaWaaeWaaeaacqaH4oqCdaahaaWcbeqaaiaaisdaaaGccq GHRaWkcaaI0aGaeqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaSIa aGymaiaaiIdacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkca aIXaGaaGOmaiabeI7aXjabgUcaRiaaigdacaaIYaaacaGLOaGaayzk aaWaaWbaaSqabeaacaaIZaGaai4laiaaikdaaaaaaaaa@6EC7@

β 2 = 3( 3 θ 8 +48 θ 7 +304 θ 6 +944 θ 5 +1816 θ 4 +2304 θ 3 +1920 θ 2 +960θ+240 ) ( θ 4 +8 θ 3 +24 θ 2 +24θ+12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIZaWaaeWaaqaa beqaaiaaiodacqaH4oqCdaahaaWcbeqaaiaaiIdaaaGccqGHRaWkca aI0aGaaGioaiabeI7aXnaaCaaaleqabaGaaG4naaaakiabgUcaRiaa iodacaaIWaGaaGinaiabeI7aXnaaCaaaleqabaGaaGOnaaaakiabgU caRiaaiMdacaaI0aGaaGinaiabeI7aXnaaCaaaleqabaGaaGynaaaa kiabgUcaRiaaigdacaaI4aGaaGymaiaaiAdacqaH4oqCdaahaaWcbe qaaiaaisdaaaaakeaacqGHRaWkcaaIYaGaaG4maiaaicdacaaI0aGa eqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGymaiaaiMdaca aIYaGaaGimaiabeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaa iMdacaaI2aGaaGimaiabeI7aXjabgUcaRiaaikdacaaI0aGaaGimaa aacaGLOaGaayzkaaaabaWaaeWaaeaacqaH4oqCdaahaaWcbeqaaiaa isdaaaGccqGHRaWkcaaI4aGaeqiUde3aaWbaaSqabeaacaaIZaaaaO Gaey4kaSIaaGOmaiaaisdacqaH4oqCdaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcaaIYaGaaGinaiabeI7aXjabgUcaRiaaigdacaaIYaaaca GLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaaaaa@8040@

β 2 = 3( 3 θ 8 +24 θ 7 +172 θ 6 +376 θ 5 +736 θ 4 +864 θ 3 +912 θ 2 +480θ+240 ) ( θ 4 +4 θ 3 +18 θ 2 +12θ+12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIZaWaaeWaaqaa beqaaiaaiodacqaH4oqCdaahaaWcbeqaaiaaiIdaaaGccqGHRaWkca aIYaGaaGinaiabeI7aXnaaCaaaleqabaGaaG4naaaakiabgUcaRiaa igdacaaI3aGaaGOmaiabeI7aXnaaCaaaleqabaGaaGOnaaaakiabgU caRiaaiodacaaI3aGaaGOnaiabeI7aXnaaCaaaleqabaGaaGynaaaa kiabgUcaRiaaiEdacaaIZaGaaGOnaiabeI7aXnaaCaaaleqabaGaaG inaaaaaOqaaiabgUcaRiaaiIdacaaI2aGaaGinaiabeI7aXnaaCaaa leqabaGaaG4maaaakiabgUcaRiaaiMdacaaIXaGaaGOmaiabeI7aXn aaCaaaleqabaGaaGOmaaaakiabgUcaRiaaisdacaaI4aGaaGimaiab eI7aXjabgUcaRiaaikdacaaI0aGaaGimaaaacaGLOaGaayzkaaaaba WaaeWaaeaacqaH4oqCdaahaaWcbeqaaiaaisdaaaGccqGHRaWkcaaI 0aGaeqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaGymaiaaiI dacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIXaGaaGOm aiabeI7aXjabgUcaRiaaigdacaaIYaaacaGLOaGaayzkaaWaaWbaaS qabeaacaaIYaaaaaaaaaa@7E10@

γ= σ 2 μ 1 = θ 4 +8 θ 3 +24 θ 2 +24θ+12 θ( θ 2 +2θ+2 )( θ 2 +4θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaSaaaeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakeaacqaH 8oqBdaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaOGamai4gkdiIc aaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWcbeqaaiaaisdaaaGc cqGHRaWkcaaI4aGaeqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaS IaaGOmaiaaisdacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWk caaIYaGaaGinaiabeI7aXjabgUcaRiaaigdacaaIYaaabaGaeqiUde 3aaeWaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI YaGaeqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaamaabmaabaGaeq iUde3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinaiabeI7aXjab gUcaRiaaiAdaaiaawIcacaGLPaaaaaaaaa@68B2@

γ= σ 2 μ 1 = θ 4 +4 θ 3 +18 θ 2 +12θ+12 θ( θ 2 +θ+2 )( θ 2 +2θ+6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaSaaaeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakeaacqaH 8oqBdaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaOGamai4gkdiIc aaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWcbeqaaiaaisdaaaGc cqGHRaWkcaaI0aGaeqiUde3aaWbaaSqabeaacaaIZaaaaOGaey4kaS IaaGymaiaaiIdacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWk caaIXaGaaGOmaiabeI7aXjabgUcaRiaaigdacaaIYaaabaGaeqiUde 3aaeWaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGccqGHRaWkcqaH 4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaeWaaeaacqaH4oqCda ahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIYaGaeqiUdeNaey4kaSIa aGOnaaGaayjkaiaawMcaaaaaaaa@67F0@

Table 1: Characteristics of Aradhana and Sujatha 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIYaaaaaGcbaGaeqiUdeNaey4kaSIaaGymaaaada qadaqaaiaaigdacqGHRaWkcaWG0baacaGLOaGaayzkaaGaamyzamaa CaaaleqabaGaeyOeI0IaeqiUdeNaaGPaVlaadshaaaaaaa@4AD2@

f( t )=θ e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iabeI7aXjaadwgadaah aaWcbeqaaiabgkHiTiabeI7aXjaaykW7caWG0baaaaaa@425D@

F( t )=1 θ+1+θt θ+1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsisldaWc aaqaaiabeI7aXjabgUcaRiaaigdacqGHRaWkcqaH4oqCcaaMc8Uaam iDaaqaaiabeI7aXjabgUcaRiaaigdaaaGaamyzamaaCaaaleqabaGa eyOeI0IaeqiUdeNaaGPaVlaadshaaaaaaa@4E01@

F( t )=1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOramaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iaaigdacqGHsislcaWG LbWaaWbaaSqabeaacqGHsislcqaH4oqCcaaMc8UaamiDaaaaaaa@422F@

h( t )= θ 2 ( 1+t ) θ+1+θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUde3a aWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIXaGaey4kaSIaamiDaa GaayjkaiaawMcaaaqaaiabeI7aXjabgUcaRiaaigdacqGHRaWkcqaH 4oqCcaaMc8UaamiDaaaaaaa@49B2@

h( t )=θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiAamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9iabeI7aXbaa@3C21@

m( t )= θ+2+θt θ( θ+1+θt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaeqiUdeNa ey4kaSIaaGOmaiabgUcaRiabeI7aXjaaykW7caWG0baabaGaeqiUde 3aaeWaaeaacqaH4oqCcqGHRaWkcaaIXaGaey4kaSIaeqiUdeNaaGPa VlaadshaaiaawIcacaGLPaaaaaaaaa@4E9E@

m( t )= 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBamaabm aabaGaamiDaaGaayjkaiaawMcaaiabg2da9maalaaabaGaaGymaaqa aiabeI7aXbaaaaa@3CF1@

μ 1 = θ+2 θ( θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOWaaWbaaSqabeaakiadacUHYaIOaaGaeyyp a0ZaaSaaaeaacqaH4oqCcqGHRaWkcaaIYaaabaGaeqiUde3aaeWaae aacqaH4oqCcqGHRaWkcaaIXaaacaGLOaGaayzkaaaaaaaa@46B8@

μ 1 = 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOWaaWbaaSqabeaakiadacUHYaIOaaGaeyyp a0ZaaSaaaeaacaaIXaaabaGaeqiUdehaaaaa@3F43@

μ 2 = θ 2 +4θ+2 θ 2 ( θ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWc beqaaiaaikdaaaGccqGHRaWkcaaI0aGaeqiUdeNaey4kaSIaaGOmaa qaaiabeI7aXnaaCaaaleqabaGaaGOmaaaakmaabmaabaGaeqiUdeNa ey4kaSIaaGymaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaa aaaa@49BF@

μ 2 = 1 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaeqiU de3aaWbaaSqabeaacaaIYaaaaaaaaaa@3D0E@

C.V= σ μ 1 = θ 2 +4θ+2 θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaac6 cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacuaH8oqBgaqbamaa BaaaleaacaaIXaaabeaaaaGccqGH9aqpdaWcaaqaamaakaaabaGaeq iUde3aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaaGinaiabeI7aXjab gUcaRiaaikdaaSqabaaakeaacqaH4oqCcqGHRaWkcaaIYaaaaaaa@4A03@

C.V= σ μ 1 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiaac6 cacaWGwbGaeyypa0ZaaSaaaeaacqaHdpWCaeaacuaH8oqBgaqbamaa BaaaleaacaaIXaaabeaaaaGccqGH9aqpcaaIXaaaaa@3F98@

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

 

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

β 2 = 3( 3 θ 4 +24 θ 3 +44 θ 2 +32θ+8 ) ( θ 2 +4θ+2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaSaaaeaacaaIZaWaaeWaaqaa beqaaiaaiodacqaH4oqCdaahaaWcbeqaaiaaisdaaaGccqGHRaWkca aIYaGaaGinaiabeI7aXnaaCaaaleqabaGaaG4maaaakiabgUcaRiaa isdacaaI0aGaeqiUde3aaWbaaSqabeaacaaIYaaaaaGcbaGaey4kaS IaaG4maiaaikdacqaH4oqCcqGHRaWkcaaI4aaaaiaawIcacaGLPaaa aeaadaqadaqaaiabeI7aXnaaCaaaleqabaGaaGOmaaaakiabgUcaRi aaisdacqaH4oqCcqGHRaWkcaaIYaaacaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaaaaaaaaaa@5925@

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

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0ZaaSaaaeaacqaHdpWCdaahaaWcbeqaaiaaikdaaaaakeaacqaH 8oqBdaWgaaWcbaGaaGymaaqabaGcdaahaaWcbeqaaOGamai4gkdiIc aaaaGaeyypa0ZaaSaaaeaacqaH4oqCdaahaaWcbeqaaiaaikdaaaGc cqGHRaWkcaaI0aGaeqiUdeNaey4kaSIaaGOmaaqaaiabeI7aXnaabm aabaGaeqiUdeNaey4kaSIaaGymaaGaayjkaiaawMcaamaabmaabaGa eqiUdeNaey4kaSIaaGOmaaGaayjkaiaawMcaaaaaaaa@5551@

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

Table 2: Characteristics of Lindley and Exponential Distributions.

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  for Aradhana Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaai4jaaaa@3949@

299.000

59.001

29.005

9.033

5.077

2.200

1.310

0.900

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaaaa@3895@

29999.990

1199.954

299.914

33.143

11.763

2.760

1.134

0.590

CV

0.579

0.587

0.597

0.637

0.676

0.755

0.813

0.853

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaaaaa@388F@

1.155

1.155

1.155

1.167

1.193

1.295

1.402

1.496

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaaaa@3880@

5.000

5.000

5.001

5.024

5.087

5.381

5.758

6.135

γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379E@

100.334

20.338

10.340

3.669

2.317

1.255

0.865

0.656

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  for Sujatha Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaai4jaaaa@3949@

299.493

59.464

29.431

9.331

5.273

2.250

1.304

0.875

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaaaa@3895@

30000.737

1200.69

300.624

33.722

12.198

2.938

1.197

0.609

CV

0.578

0.583

0.589

0.622

0.662

0.762

0.839

0.892

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaaaaa@388F@

1.155

1.154

1.151

1.140

1.146

1.248

1.397

1.536

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaaaa@3880@

5.000

4.998

4.992

4.955

4.945

5.170

5.656

6.215

γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379E@

100.172

20.192

10.214

3.614

2.313

1.306

0.918

0.696

Values of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  for Lindley Distribution

0.01

0.05

0.1

0.3

0.5

1

1.5

2

μ 1 ' MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOGaai4jaaaa@3949@

199.010

39.048

19.091

5.897

3.333

1.500

0.933

0.667

μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaaaa@3895@

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@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaaaaa@388F@

1.414

1.417

1.422

1.464

1.512

1.620

1.699

1.756

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaaaa@3880@

6.000

6.007

6.025

6.162

6.343

6.796

7.173

7.469

γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379E@

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaigdaaeqaaOWaaWbaaSqabeaakiadacUHYaIOaaaaaa@3BBC@ , μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS baaSqaaiaaikdaaeqaaaaa@3894@ ,C.V, β 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaaaaa@388F@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaaaa@3880@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379E@  of Aradhana, Sujatha and Lindley distributions for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@ .

SD: Standard Deviation; BMI: Body Mass Index; WC: Waist Circumference; AC: Abdominal Circumference; HC: Hip Circumference; RER: Respiratory Exchange Ratio; HR: Hear Rate.

Distribution

Over-Dispersion ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH8aapcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CEF@

Equi-Dispersion ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH9aqpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF1@

Under-Dispersion ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aH8oqBcqGH+aGpcqaHdpWCdaahaaWcbeqaaiaaikdaaaaakiaawIca caGLPaaaaaa@3CF3@

Aradhana

θ<1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ipaWJaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOmaiaa iAdacaaI1aGaaGimaiaaiwdaaaa@40CE@

θ=1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0JaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOmaiaa iAdacaaI1aGaaGimaiaaiwdaaaa@40D0@

θ>1.283826505 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey Opa4JaaGymaiaac6cacaaIYaGaaGioaiaaiodacaaI4aGaaGOmaiaa iAdacaaI1aGaaGimaiaaiwdaaaa@40D2@

Sujatha

θ<1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ipaWJaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4naiaa igdacaaIXaGaaG4naiaaisdaaaa@40CA@

θ=1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0JaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4naiaa igdacaaIXaGaaG4naiaaisdaaaa@40CC@

θ>1.364271174 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey Opa4JaaGymaiaac6cacaaIZaGaaGOnaiaaisdacaaIYaGaaG4naiaa igdacaaIXaGaaG4naiaaisdaaaa@40CE@

Lindley

θ<1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ipaWJaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGioaiaa iAdacaaI0aGaaGioaiaaiEdaaaa@40D0@

θ=1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGioaiaa iAdacaaI0aGaaGioaiaaiEdaaaa@40D2@

θ>1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey Opa4JaaGymaiaac6cacaaIXaGaaG4naiaaicdacaaIWaGaaGioaiaa iAdacaaI0aGaaGioaiaaiEdaaaa@40D4@

Exponential

θ<1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ipaWJaaGymaaaa@396B@

θ=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey ypa0JaaGymaaaa@396D@

θ>1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdeNaey Opa4JaaGymaaaa@396F@

Table 4: Over-dispersion, equi-dispersion and under-dispersion of Aradhana, Sujatha, 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 coefficient of variation (C.V), coefficient of skewness ( β 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGydaWgaaWcbaGaaGymaaqabaaabeaaaaa@388F@ ) coefficient of kurtosis ( β 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aaS baaSqaaiaaikdaaeqaaaaa@3880@ ) and index of dispersion ( γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCgaaa@379E@ ) for Aradhana, Sujatha, and Lindley distributions are for varying values of their parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AC@ .

Model

Parameter Estimate

-2ln L

AIC

AICC

BIC

K-S Statistic

Data 1

Aradhana

1.346393

149.88

151.88

151.94

154.02

0.345

Sujatha

1.350050

154.81

156.81

156.87

158.95

0.349

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

Aradhana

0.043272

952.58

954.58

954.62

957.18

0.186

Sujatha

0.043566

951.78

953.78

953.97

954.91

0.185

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

Aradhana

0.040968

227.28

229.28

229.47

230.41

0.108

Sujatha

0.041232

227.17

229.17

229.36

230.30

0.107

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

Aradhana

0.013454

1255.26

1257.26

1257.30

1259.86

0.069

Sujatha

0.013484

1255.54

1257.54

1257.58

1260.14

0.070

Lindley

0.00897

1251.34

1253.34

1253.38

1255.95

0.098

Exponential

0.004505

1280.52

1282.52

1282.56

1285.12

0.190

Data 5

Aradhana

0.029756

794.28

796.28

796.34

798.56

0.182

Sujatha

0.029898

794.48

796.48

796.54

798.77

0.183

Lindley

0.019841

789.04

791.04

791.10

793.32

0.133

Exponential

0.010018

806.88

808.88

808.94

811.16

0.198

Data 6

Aradhana

0.115577

989.49

991.49

991.52

994.39

0.399

Sujatha

0.117453

985.69

987.69

987.72

990.59

0.396

Lindley

0.077247

1041.64

1043.64

1043.68

1046.54

0.448

Exponential

0.04006

1130.26

1132.26

1132.29

1135.16

0.525

Data 7

Aradhana

0.013206

801.83

803.83

803.90

805.89

0.297

Sujatha

0.013234

802.84

804.84

804.91

806.90

0.298

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

Aradhana

0.013364

608.87

610.87

610.96

612.65

0.278

Sujatha

0.013394

609.39

611.39

611.48

613.17

0.279

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

Aradhana

0.290304

874.71

876.71

876.74

879.56

0.179

Sujatha

0.298963

879.82

881.82

881.85

884.67

0.187

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

Aradhana

0.049506

350.55

352.55

352.69

353.95

0.415

Sujatha

0.049887

352.47

354.47

354.61

355.87

0.418

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

Aradhana

1.132874

116.06

118.06

118.18

119.59

0.169

Sujatha

1.146073

115.54

117.54

117.66

119.07

0.164

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

Aradhana

0.276551

638.34

640.34

640.38

642.94

0.080

Sujatha

0.284621

639.64

641.64

641.68

644.24

0.088

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

Aradhana

0.024537

193.60

195.60

195.91

196.31

0.453

Sujatha

0.024634

193.94

195.94

196.25

196.65

0.454

Lindley

0.01636

181.34

183.34

183.65

184.05

0.386

Exponential

0.008246

173.94

175.94

176.25

176.65

0.277

Data 14

Aradhana

1.123193

56.37

58.37

58.59

59.36

0.302

Sujatha

1.136745

57.50

59.50

59.72

60.49

0.309

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

Aradhana

0.094318

242.23

244.23

244.37

245.66

0.274

Sujatha

0.095610

241.50

243.50

243.64

244.93

0.270

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

Aradhana

0.917023

219.90

221.90

221.96

224.13

0.350

Sujatha

0.936119

221.61

223.61

223.67

225.84

0.362

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 [22].

5

25

31

32

34

35

38

39

39

40

42

43

43

43

44

44

47

47

48

49

49

49

51

54

55

55

55

56

56

56

58

59

59

59

59

59

63

63

64

64

65

65

65

66

66

66

66

66

67

67

67

68

69

69

69

69

71

71

72

73

73

73

74

74

76

76

77

77

77

77

77

77

79

79

80

81

83

83

84

86

86

87

90

91

92

92

92

92

93

94

97

98

98

99

101

103

105

109

136

147

Data Set 2: The data is given by Birnbaum & Saunders [23] on the fatigue life of 6061 – T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data ( × MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqlaaa@380D@ ) 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 [24]. 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 [25] 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 guinea pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal [26].

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 et al. [27]: 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

16.1

22.7

34

41.55

42

45.28

49.4

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 [28] represent the survival times of a group of patients suffering from Head and Neck cancer disease and treated using radiotherapy (RT

12.20

23.56

23.74

25.87

31.98

37

41.35

47.38

55.46

58.36

63.47

68.46

78.26

74.47

81.43

84

92

94

110

112

119

127

130

133

140

146

155

159

173

179

194

195

209

249

281

319

339

432

469

519

633

725

817

1776

Data Set 8: The data set reported by Efron [28] 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

18.1

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 [29].

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 [30], 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. [31], is vinyl chloride data obtained from clean upgradient monitoring wells in mg/l.

0.8,

0.8,

1.3,

1.5,

1.8,

1.9,

1.9,

2.1,

2.6,

2.7,

2.9,

3.1,

3.2,

3.3,

3.5,

3.6,

4.0,

4.1,

4.2,

4.2,

4.3,

4.3,

4.4,

4.4,

4.6,

4.7,

4.7,

4.8,

4.9,

4.9,

5.0,

5.3,

5.5,

5.7,

5.7,

6.1,

6.2,

6.2,

6.2,

6.3,

6.7,

6.9,

7.1,

7.1,

7.1,

7.1,

7.4,

7.6,

7.7,

8.0,

8.2,

8.6,

8.6,

8.6,

8.8,

8.8,

8.9,

8.9,

9.5,

9.6,

9.7,

9.8,

10.7,

10.9,

11.0,

11.0,

11.1,

11.2,

11.2,

11.5,

11.9,

12.4,

12.5,

12.9,

13.0,

13.1,

13.3,

13.6,

13.7,

13.9,

14.1,

15.4,

15.4,

17.3,

17.3,

18.1,

18.2,

18.4,

18.9,

19.0,

19.9,

20.6,

21.3,

21.4,

21.9,

23.0,

27.0,

31.6,

33.1,

38.5

Data set 12: This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al. [7] for fitting the Lindley [6] 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 [32].

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 [33].

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. [34].

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 [35].

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