ISSN: 2378-315X BBIJ

Biometrics & Biostatistics International Journal
Research Article
Volume 3 Issue 2 - 2016
On Modeling of Lifetime Data Using One Parameter Akash, Lindley and Exponential Distributions
Rama Shanker1*, Hagos Fesshaye2, Sujatha Selvaraj3
1Department of Statistics, Eritrea Institute of Technology, Eritrea
2Department of Economics, College of Business and Economics, Eritrea
3Department of Banking and Finance, Jimma University, Ethiopia
Received: December 02, 2015 | Published: January 28, 2016
*Corresponding author: Rama Shanker, Department of Statistics, Eritrea Institute of Technology, Asmara, Eritrea, Email:
Citation: Shanker R, Fesshaye H, Selvaraj S (2016) On Modeling of Lifetime Data Using One Parameter Akash, Lindley and Exponential Distributions. Biom Biostat Int J 3(2): 00061. DOI: 10.15406/bbij.2016.03.00061

Abstract

The analysis and modeling of lifetime data are crucial in almost all applied sciences including medicine, insurance, engineering, and finance, amongst others. In the present paper an attempt has been made to discuss applications of Akash distribution introduced by Shanker [1], Lindley distribution and exponential distributions for modeling lifetime data from various fields. Firstly a table for values of the various characteristics of Akash distribution and Lindley distribution has been presented for various values of their parameter which reflects their nature and behavior. The expressions for the index of dispersion of Akash, Lindley and exponential distributions have been obtained and the conditions under which Akash, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed has been given. Several lifetime data from medical science and engineering have been fitted using Akash distribution along with Lindley and exponential distributions to study the advantages and disadvantages of these distributions for modeling lifetime data.

Keywords: Akash distribution; Lindley distribution; Exponential distribution; Index of dispersion; 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.

Recently Shanker [1] has introduced a one parameter continuous distribution named, “Akash 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 Akash, Lindley and exponential distributions are of one parameter, Akash and Lindley distributions have advantage over the exponential distribution that the exponential distribution has constant hazard rate and mean residual life function whereas the Akash and Lindley distributions have increasing hazard rate and decreasing mean residual life function. Further, Akash distribution of Shanker [1] has flexibility over both Lindley and exponential distributions.

Exponential, Lindley and Akash Distributions

  1. Exponential distribution
  2. In statistical literature, exponential distribution was the first widely used lifetime distribution model in areas ranging from studies on the lifetimes of manufactured items Davis [2], Epstein & Sobel [3], Epstein [4] to research involving survival or remission times in chronic diseases Feigl & Zelen [5]. The main reason for its wide usefulness and applicability as lifetime model is partly because of the availability of simple statistical methods for it Epstein & Sobel [3] and partly because it appeared suitable for representing the lifetimes of many things such as various types of manufactured items Davis [2].

  3. Lindley distribution
  4. The Lindley distribution is a two-component mixture of an exponential distribution having scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  and a gamma distribution having shape parameter 2 and scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  with mixing proportions θ θ+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeqiUdehakeaajugibiabeI7aXjabgUcaRiaaigdaaaaa aa@3CCF@ and 1 θ+1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGymaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaaaaaaa @3BD4@  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, modified and its detailed applications in reliability and other fields of knowledge by different researchers including Hussain [8], Zakerzadeh & Dolati [9], Nadarajah et al. [10], Deniz & Ojeda [11], Bakouch et al. [12], Shanker & Mishra [13,14], Shanker et al. [15], Elbatal et al. [16], Ghitany et al. [17], Merovci [18], Liyanage & Pararai [19], Ashour & Eltehiwy [20], Oluyede & Yang [21], Singh et al. [22], Sharma et al. [23], Shanker et al. [24], Alkarni [25], Pararai et al. [26], Abouammoh et al. [27] are some among others.

    Although the Lindley distribution has been used to model lifetime data by many researchers and Hussain [8] has shown that the Lindley distribution is important for studying stress-strength reliability modeling, it has been observed that there are many situations in the modeling of lifetime data where the Lindley distribution may not be suitable from a theoretical or applied point of view. In fact, Shanker et al. [24] has detailed comparative study about the applicability of Lindley and exponential distributions for modeling various types of lifetime data and observed that none is a suitable model in all cases.

  5. Akash distribution

Shanker [1] introduced a new distribution named, ‘Akash distribution’ which is flexible than the Lindley distribution for modeling lifetime data in reliability and in terms of its hazard rate shapes. Akash distribution is a two- component mixture of an exponential distribution having scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  and a gamma distribution having shape parameter 3 and scale parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@  with mixing proportions θ 2 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikdaaaaakeaa jugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey 4kaSIaaGOmaaaaaaa@42A9@ and 1 θ 2 +2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaaGymaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqz adGaaGOmaaaajugibiabgUcaRiaaikdaaaaaaa@3F09@ and has been shown by Shanker [1] that Akash distribution gives better fit than Lindley and exponential distributions in modeling lifetime data.

Let T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGub aaaa@375E@ 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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaaaa@3B23@  , cumulative distribution function, F( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaaaa@3B03@  , survival function, S( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaaaa@3B10@ , hazard rate function, h( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaaaa@3B25@ , mean residual life function, m( t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaaaaa@3B2A@ , mean μ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaaaa@3FBA@ , variance μ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaaaa@3AEA@ , third moment about mean μ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIZaaaleqaaaaa@3AEB@  , fourth moment about mean μ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaI0aaaleqaaaaa@3AEC@ , coefficient of variation (C.V.), coefficient of Skewness ( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaakaaakeaajugibiabek7aILqbaoaaBaaaleaajugWaiaa igdaaSqabaaabeaaaOGaayjkaiaawMcaaaaa@3DA7@ , coefficient of Kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaOGa ayjkaiaawMcaaaaa@3D00@ , and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeq4SdCgakiaawIcacaGLPaaaaaa@3A57@  of exponential, Lindley and Akash distributions are summarized in the following .

Exponential Distribution

Lindley Distribution

Akash Distribution

f( t )=θ e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcqaH4oqCcaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiabeI 7aXjaaykW7caWG0baaaaaa@4668@

f( t )= θ 2 θ+1 ( 1+t ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaGOmaaaaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGymaaaajuaGdaqa daGcbaqcLbsacaaIXaGaey4kaSIaamiDaaGccaGLOaGaayzkaaqcLb sacaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiabeI7aXjaaykW7 caWG0baaaaaa@540F@

f( t )= θ 3 θ 2 +2 ( 1+ t 2 ) e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGMb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaG4maaaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaa ikdaaaqcLbsacqGHRaWkcaaIYaaaaKqbaoaabmaakeaajugibiaaig dacqGHRaWkcaWG0bqcfa4aaWbaaSqabeaajugWaiaaikdaaaaakiaa wIcacaGLPaaajugibiaadwgajuaGdaahaaWcbeqaaKqzadGaeyOeI0 IaeqiUdeNaaGPaVlaadshaaaaaaa@59EA@

F( t )=1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0IaamyzaKqbaoaaCaaaleqabaqcLbmacqGHsi slcqaH4oqCcaaMc8UaamiDaaaaaaa@463A@

F( t )=1 θ+1+θt θ+1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqiUdeNaey4kaS IaaGymaiabgUcaRiabeI7aXjaaykW7caWG0baakeaajugibiabeI7a XjabgUcaRiaaigdaaaGaamyzaKqbaoaaCaaaleqabaqcLbmacqGHsi slcqaH4oqCcaaMc8UaamiDaaaaaaa@53CC@

F( t )=1[ 1+ θt( θt+2 ) θ 2 +2 ] e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGgb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIXaGaeyOeI0scfa4aamWaaOqaaKqzGeGaaGymaiabgUcaRK qbaoaalaaakeaajugibiabeI7aXjaaykW7caWG0bqcfa4aaeWaaOqa aKqzGeGaeqiUdeNaaGPaVlaadshacqGHRaWkcaaIYaaakiaawIcaca GLPaaaaeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaa aKqzGeGaey4kaSIaaGOmaaaaaOGaay5waiaaw2faaKqzGeGaamyzaK qbaoaaCaaaleqabaqcLbmacqGHsislcqaH4oqCcaaMc8UaamiDaaaa aaa@60A3@

S( t )= e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiabeI7aXjaayk W7caWG0baaaaaa@449F@

S( t )= θ+1+θt θ+1 e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIXaGaey4kaS IaeqiUdeNaaGPaVlaadshaaOqaaKqzGeGaeqiUdeNaey4kaSIaaGym aaaacaWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiabeI7aXjaayk W7caWG0baaaaaa@5231@

S( t )=[ 1+ θt( θt+2 ) θ 2 +2 ] e θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGtb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWadaGcbaqcLbsacaaIXaGaey4kaSscfa4aaSaaaOqaaK qzGeGaeqiUdeNaaGPaVlaadshajuaGdaqadaGcbaqcLbsacqaH4oqC caaMc8UaamiDaiabgUcaRiaaikdaaOGaayjkaiaawMcaaaqaaKqzGe GaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqGHRaWk caaIYaaaaaGccaGLBbGaayzxaaqcLbsacaWGLbqcfa4aaWbaaSqabe aajugWaiabgkHiTiabeI7aXjaaykW7caWG0baaaaaa@5F08@

h( t )=θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcqaH4oqCaaa@3E70@

h( t )= θ 2 ( 1+t ) θ+1+θt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaGOmaaaajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIaamiDaaGc caGLOaGaayzkaaaabaqcLbsacqaH4oqCcqGHRaWkcaaIXaGaey4kaS IaeqiUdeNaaGPaVlaadshaaaaaaa@509A@

h( t )= θ 3 ( 1+ t 2 ) θt( θt+2 )+( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGOb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaG4maaaajuaGdaqadaGcbaqcLbsacaaIXaGaey4kaSIaamiDaKqb aoaaCaaaleqabaqcLbmacaaIYaaaaaGccaGLOaGaayzkaaaabaqcLb sacqaH4oqCcaaMc8UaamiDaKqbaoaabmaakeaajugibiabeI7aXjaa ykW7caWG0bGaey4kaSIaaGOmaaGccaGLOaGaayzkaaqcLbsacqGHRa WkjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGa aGOmaaaajugibiabgUcaRiaaikdaaOGaayjkaiaawMcaaaaaaaa@6250@

m( t )= 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiabeI7aXbaaaa a@4100@

m( t )= θ+2+θt θ( θ+1+θt ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCcqGHRaWkcaaIYaGaey4kaS IaeqiUdeNaaGPaVlaadshaaOqaaKqzGeGaeqiUdexcfa4aaeWaaOqa aKqzGeGaeqiUdeNaey4kaSIaaGymaiabgUcaRiabeI7aXjaaykW7ca WG0baakiaawIcacaGLPaaaaaaaaa@53DE@

m( t )= θ 2 t 2 +4θt+( θ 2 +6 ) θ[ θt( θt+2 )+( θ 2 +2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGTb qcfa4aaeWaaOqaaKqzGeGaamiDaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaGOmaaaajugibiaaykW7caWG0bqcfa4aaWbaaSqabeaajugWaiaa ikdaaaqcLbsacqGHRaWkcaaI0aGaeqiUdeNaaGPaVlaadshacqGHRa WkjuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGa aGOmaaaajugibiabgUcaRiaaiAdaaOGaayjkaiaawMcaaaqaaKqzGe GaeqiUdexcfa4aamWaaOqaaKqzGeGaeqiUdeNaaGPaVlaadshajuaG daqadaGcbaqcLbsacqaH4oqCcaaMc8UaamiDaiabgUcaRiaaikdaaO GaayjkaiaawMcaaKqzGeGaey4kaSscfa4aaeWaaOqaaKqzGeGaeqiU dexcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIYa aakiaawIcacaGLPaaaaiaawUfacaGLDbaaaaaaaa@7577@

μ 1 = 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaG ymaaGcbaqcLbsacqaH4oqCaaaaaa@4590@

μ 1 = θ+2 θ( θ+1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeq iUdeNaey4kaSIaaGOmaaGcbaqcLbsacqaH4oqCjuaGdaqadaGcbaqc LbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaaaaaaaa@4E36@

μ 1 = θ 2 +6 θ( θ 2 +2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaKqbaoaaCaaaleqabaqc LbmacWaGGBOmGikaaKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaeq iUdexcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaI 2aaakeaajugibiabeI7aXLqbaoaabmaakeaajugibiabeI7aXLqbao aaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOmaaGccaGL OaGaayzkaaaaaaaa@54A3@

μ 2 = 1 θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbe qaaKqzadGaaGOmaaaaaaaaaa@4365@

μ 2 = θ 2 +4θ+2 θ 2 ( θ+1 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikdaaa qcLbsacqGHRaWkcaaI0aGaeqiUdeNaey4kaSIaaGOmaaGcbaqcLbsa cqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajuaGdaqadaGcba qcLbsacqaH4oqCcqGHRaWkcaaIXaaakiaawIcacaGLPaaajuaGdaah aaWcbeqaaKqzadGaaGOmaaaaaaaaaa@553A@

μ 2 = θ 4 +16 θ 2 +12 θ 2 ( θ 2 +2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaisdaaa qcLbsacqGHRaWkcaaIXaGaaGOnaiabeI7aXLqbaoaaCaaaleqabaqc LbmacaaIYaaaaKqzGeGaey4kaSIaaGymaiaaikdaaOqaaKqzGeGaeq iUdexcfa4aaWbaaSqabeaajugWaiaaikdaaaqcfa4aaeWaaOqaaKqz GeGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqGHRa WkcaaIYaaakiaawIcacaGLPaaajuaGdaahaaWcbeqaaKqzadGaaGOm aaaaaaaaaa@5D1D@

μ 3 = 2 θ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIZaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaGOmaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbe qaaKqzadGaaG4maaaaaaaaaa@4368@

μ 3 = 2( θ 3 +6 θ 2 +6θ+2 ) θ 3 ( θ+1 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIZaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaGOmaKqbaoaabmaakeaajugibiabeI7aXLqbao aaCaaaleqabaqcLbmacaaIZaaaaKqzGeGaey4kaSIaaGOnaiabeI7a XLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOnai abeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaaqaaKqzGeGaeqiU dexcfa4aaWbaaSqabeaajugWaiaaiodaaaqcfa4aaeWaaOqaaKqzGe GaeqiUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqa beaajugWaiaaiodaaaaaaaaa@5F38@

μ 3 = 2( θ 6 +30 θ 4 +36 θ 2 +24 ) θ 3 ( θ 2 +2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaIZaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaGOmaKqbaoaabmaakeaajugibiabeI7aXLqbao aaCaaaleqabaqcLbmacaaI2aaaaKqzGeGaey4kaSIaaG4maiaaicda cqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGinaaaajugibiabgUcaRi aaiodacaaI2aGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaikdaaaqc LbsacqGHRaWkcaaIYaGaaGinaaGccaGLOaGaayzkaaaabaqcLbsacq aH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4maaaajuaGdaqadaGcbaqc LbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiabgU caRiaaikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLbmacaaI Zaaaaaaaaaa@67D8@

μ 4 = 9 θ 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaI0aaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaGyoaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbe qaaKqzadGaaGinaaaaaaaaaa@4371@

μ 4 = 3( 3 θ 4 +24 θ 3 +44 θ 2 +32θ+8 ) θ 4 ( θ+1 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaI0aaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaG4maKqbaoaabmaajugibqaabeGcbaqcLbsaca aIZaGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaisdaaaqcLbsacqGH RaWkcaaIYaGaaGinaiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZa aaaKqzGeGaey4kaSIaaGinaiaaisdacqaH4oqCjuaGdaahaaWcbeqa aKqzadGaaGOmaaaaaOqaaKqzGeGaey4kaSIaaG4maiaaikdacqaH4o qCcqGHRaWkcaaI4aaaaOGaayjkaiaawMcaaaqaaKqzGeGaeqiUdexc fa4aaWbaaSqabeaajugWaiaaisdaaaqcfa4aaeWaaOqaaKqzGeGaeq iUdeNaey4kaSIaaGymaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaa jugWaiaaisdaaaaaaaaa@695C@

μ 4 = 3( 3 θ 8 +128 θ 6 +408 θ 4 +576 θ 2 +240 ) θ 4 ( θ 2 +2 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH8o qBjuaGdaWgaaWcbaqcLbmacaaI0aaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaG4maKqbaoaabmaajugibqaabeGcbaqcLbsaca aIZaGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaiIdaaaqcLbsacqGH RaWkcaaIXaGaaGOmaiaaiIdacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaGOnaaaajugibiabgUcaRiaaisdacaaIWaGaaGioaiabeI7aXLqb aoaaCaaaleqabaqcLbmacaaI0aaaaaGcbaqcLbsacqGHRaWkcaaI1a GaaG4naiaaiAdacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaa jugibiabgUcaRiaaikdacaaI0aGaaGimaaaakiaawIcacaGLPaaaae aajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaI0aaaaKqbaoaa bmaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaK qzGeGaey4kaSIaaGOmaaGccaGLOaGaayzkaaqcfa4aaWbaaSqabeaa jugWaiaaisdaaaaaaaaa@7384@

C.V= σ μ 1 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb GaaiOlaiaadAfacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCaOqa aKqzGeGafqiVd0MbauaajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaa aajugibiabg2da9iaaigdaaaa@4433@

C.V= σ μ 1 = θ 2 +4θ+2 θ+2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb GaaiOlaiaadAfacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCaOqa aKqzGeGafqiVd0MbauaajuaGdaWgaaWcbaqcLbmacaaIXaaaleqaaa aajugibiabg2da9KqbaoaalaaakeaajuaGdaGcaaGcbaqcLbsacqaH 4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaajugibiabgUcaRiaais dacqaH4oqCcqGHRaWkcaaIYaaaleqaaaGcbaqcLbsacqaH4oqCcqGH RaWkcaaIYaaaaaaa@532D@

C.V= σ μ 1 = θ 4 +16 θ 2 +12 θ 2 +6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGdb GaaiOlaiaadAfacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCaOqa aKqzGeGaeqiVd0wcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajuaGda ahaaWcbeqaaKqzadGamai4gkdiIcaaaaqcLbsacqGH9aqpjuaGdaWc aaGcbaqcfa4aaOaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaaju gWaiaaisdaaaqcLbsacqGHRaWkcaaIXaGaaGOnaiabeI7aXLqbaoaa CaaaleqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGymaiaaikdaaS qabaaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaa aKqzGeGaey4kaSIaaGOnaaaaaaa@5FD8@

 

β 1 =2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaeqa aKqzGeGaeyypa0JaaGOmaaaa@3DCD@

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaeqa aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmaKqbaoaabmaake aajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZaaaaKqzGeGa ey4kaSIaaGOnaiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaK qzGeGaey4kaSIaaGOnaiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaa wMcaaaqaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqaba qcLbmacaaIYaaaaKqzGeGaey4kaSIaaGinaiabeI7aXjabgUcaRiaa ikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaWaaSGbaeaajugWai aaiodaaSqaaKqzadGaaGOmaaaaaaaaaaaa@6374@

β 1 = 2( θ 6 +30 θ 4 +36 θ 2 +24 ) ( θ 4 +16 θ 2 +12 ) 3/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaeqa aKqzGeGaeyypa0tcfa4aaSaaaOqaaKqzGeGaaGOmaKqbaoaabmaake aajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaI2aaaaKqzGeGa ey4kaSIaaG4maiaaicdacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG inaaaajugibiabgUcaRiaaiodacaaI2aGaeqiUdexcfa4aaWbaaSqa beaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIYaGaaGinaaGccaGLOa Gaayzkaaaabaqcfa4aaeWaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqa beaajugWaiaaisdaaaqcLbsacqGHRaWkcaaIXaGaaGOnaiabeI7aXL qbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGymaiaa ikdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqabaqcLbmacaaIZaGaai 4laiaaikdaaaaaaaaa@6CF1@

 

β 2 =9 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0JaaGyo aaaa@3D2D@

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaG4maKqbaoaabmaajugibqaabeGcbaqcLbsaca aIZaGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaisdaaaqcLbsacqGH RaWkcaaIYaGaaGinaiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIZa aaaKqzGeGaey4kaSIaaGinaiaaisdacqaH4oqCjuaGdaahaaWcbeqa aKqzadGaaGOmaaaaaOqaaKqzGeGaey4kaSIaaG4maiaaikdacqaH4o qCcqGHRaWkcaaI4aaaaOGaayjkaiaawMcaaaqaaKqbaoaabmaakeaa jugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey 4kaSIaaGinaiabeI7aXjabgUcaRiaaikdaaOGaayjkaiaawMcaaKqb aoaaCaaaleqabaqcLbmacaaIYaaaaaaaaaa@6AE2@

β 2 = 3( 3 θ 8 +128 θ 6 +408 θ 4 +576 θ 2 +240 ) ( θ 4 +16 θ 2 +12 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaKqzGeGaeyypa0tcfa4a aSaaaOqaaKqzGeGaaG4maKqbaoaabmaajugibqaabeGcbaqcLbsaca aIZaGaeqiUdexcfa4aaWbaaSqabeaajugWaiaaiIdaaaqcLbsacqGH RaWkcaaIXaGaaGOmaiaaiIdacqaH4oqCjuaGdaahaaWcbeqaaKqzad GaaGOnaaaajugibiabgUcaRiaaisdacaaIWaGaaGioaiabeI7aXLqb aoaaCaaaleqabaqcLbmacaaI0aaaaaGcbaqcLbsacqGHRaWkcaaI1a GaaG4naiaaiAdacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaGOmaaaa jugibiabgUcaRiaaikdacaaI0aGaaGimaaaakiaawIcacaGLPaaaae aajuaGdaqadaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbeqaaKqzadGa aGinaaaajugibiabgUcaRiaaigdacaaI2aGaeqiUdexcfa4aaWbaaS qabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIXaGaaGOmaaGccaGL OaGaayzkaaqcfa4aaWbaaSqabeaajugWaiaaikdaaaaaaaaa@7683@

γ= σ 2 μ 1 = 1 θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCjuaGdaahaaWcbeqa aKqzadGaaGOmaaaaaOqaaKqzGeGaeqiVd0wcfa4aaSbaaSqaaKqzad GaaGymaaWcbeaajuaGdaahaaWcbeqaaKqzadGamai4gkdiIcaaaaqc LbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiabeI 7aXbaaaaa@4E75@

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

γ= σ 2 μ 1 = θ 4 +16 θ 2 +12 θ( θ 2 +2 )( θ 2 +6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo WzcqGH9aqpjuaGdaWcaaGcbaqcLbsacqaHdpWCjuaGdaahaaWcbeqa aKqzadGaaGOmaaaaaOqaaKqzGeGaeqiVd0wcfa4aaSbaaSqaaKqzad GaaGymaaWcbeaajuaGdaahaaWcbeqaaKqzadGamai4gkdiIcaaaaqc LbsacqGH9aqpjuaGdaWcaaGcbaqcLbsacqaH4oqCjuaGdaahaaWcbe qaaKqzadGaaGinaaaajugibiabgUcaRiaaigdacaaI2aGaeqiUdexc fa4aaWbaaSqabeaajugWaiaaikdaaaqcLbsacqGHRaWkcaaIXaGaaG OmaaGcbaqcLbsacqaH4oqCjuaGdaqadaGcbaqcLbsacqaH4oqCjuaG daahaaWcbeqaaKqzadGaaGOmaaaajugibiabgUcaRiaaikdaaOGaay jkaiaawMcaaKqbaoaabmaakeaajugibiabeI7aXLqbaoaaCaaaleqa baqcLbmacaaIYaaaaKqzGeGaey4kaSIaaGOnaaGccaGLOaGaayzkaa aaaaaa@6ECE@

Table 1: Characteristics of Exponential, Lindley and Akash Distributions.

It can be easily verified that the Akash distribution is over- dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0MaeyipaWJaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD2@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0Maeyypa0Jaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD4@  and under-dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0MaeyOpa4Jaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD6@  for θ<(=)> θ =1.515400063 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcqGH8aapcaGGOaGaeyypa0Jaaiykaiabg6da+iabeI7aXLqbaoaa CaaaleqabaqcLbmacqGHxiIkaaqcLbsacqGH9aqpcaaIXaGaaiOlai aaiwdacaaIXaGaaGynaiaaisdacaaIWaGaaGimaiaaicdacaaI2aGa aG4maaaa@4AD8@  respectively. Further, Lindley distribution is over- dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0MaeyipaWJaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD2@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0Maeyypa0Jaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD4@  and under-dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0MaeyOpa4Jaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD6@  for θ<(=)> θ =1.170086487 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcqGH8aapcaGGOaGaeyypa0Jaaiykaiabg6da+iabeI7aXLqbaoaa CaaaleqabaqcLbmacqGHxiIkaaqcLbsacqGH9aqpcaaIXaGaaiOlai aaigdacaaI3aGaaGimaiaaicdacaaI4aGaaGOnaiaaisdacaaI4aGa aG4naaaa@4AE9@  respectively, whereas as exponential distribution is over- dispersed ( μ< σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0MaeyipaWJaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD2@ , equi-dispersed ( μ= σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0Maeyypa0Jaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD4@  and under- dispersed ( μ> σ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqiVd0MaeyOpa4Jaeq4Wdmxcfa4aaWbaaSqabeaajugW aiaaikdaaaaakiaawIcacaGLPaaaaaa@3FD6@  for θ<(=)> θ =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCcqGH8aapcaGGOaGaeyypa0Jaaiykaiabg6da+iabeI7aXLqbaoaa CaaaleqabaqcLbmacqGHxiIkaaqcLbsacqGH9aqpcaaIXaaaaa@4384@  respectively.

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=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqbaoaakaaakeaajugibiabek7aILqbaoaaBaaaleaajugWaiaa igdaaSqabaaabeaaaOGaayjkaiaawMcaaaaa@3DA7@ , coefficient of Kurtosis ( β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaOGa ayjkaiaawMcaaaaa@3D00@ , and index of dispersion ( γ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaaO qaaKqzGeGaeq4SdCgakiaawIcacaGLPaaaaaa@3A57@  for Akash and Lindley distributions for various values of their parameter for comparative study are summarized in the Table 2.

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

 

0.01

0.05

0.09

0.5

0.8

1.5

2

C.V

0.577379

0.578071

0.579679

0.641249

0.716741

0.882958

0.959166

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaeqa aaaa@3B7C@

1.154643

1.153268

1.150133

1.083974

1.10564

1.388077

1.61372

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaaaa@3AD5@

4.999867

4.996681

4.989352

4.784948

4.735717

5.472724

6.391304

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo Wzaaa@382C@

100.0067

20.03328

11.17079

2.284444

1.615097

1.008913

0.766667

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

 

0.01

0.05

0.09

0.5

0.8

1.5

2

C.V

0.710607

0.723943

0.736298

0.824621

0.863075

0.914732

0.935414

β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaOaaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaaaeqa aaaa@3B7C@

1.414317

1.416546

1.421076

1.512281

1.580387

1.698866

1.756288

β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHYo GyjuaGdaWgaaWcbaqcLbmacaaIYaaaleqaaaaa@3AD5@

6.000294

6.006807

6.020488

6.342561

6.621505

7.172516

7.469388

γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHZo Wzaaa@382C@

100.4926

20.46458

11.55007

2.266667

1.448413

0.780952

0.583333

Table 2: Values of C.V β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqabeqabmGadiqaceqabeqadeqabqqaaO qaaKqbaoaakaaakeaajugibiabek7aILqbaoaaBaaaleaajugWaiaa igdaaSqabaaabeaaaaa@3FB1@ , β 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqabeqabmGadiqaceqabeqadeqabqqaaO qaaKqzGeGaeqOSdiwcfa4aaSbaaSqaaKqzadGaaGOmaaWcbeaaaaa@3F0A@ and γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D aebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0x Xdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs 0dXdbPYxe9vr0=vr0=vqpWqabeqabmGadiqaceqabeqadeqabqqaaO qaaKqzGeGaeq4SdCgaaa@3C61@ of Akash and Lindley Distributions for varying values of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ .

Applications

The Akash, Lindley and exponential distributions have been fitted to a number of real lifetime data - sets to tests their goodness of fit. Goodness of fit tests for sixteen real lifetime data- sets have been presented here. In order to compare Akash, Lindley and exponential distributions, 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics ( Kolmogorov-Smirnov Statistics) for all sixteen real lifetime data- sets have been computed Table 3. The formulae for computing AIC, AICC, BIC, and K-S Statistics are as follows:

 

Model

Parameter
Estimate

-2ln L

AIC

AICC

BIC

K-S
Statistic

Data 1

Akash
Lindley

1.355445
0.996116

163.73
162.56

165.73
164.56

165.79
164.62

169.93
166.70

0.355
0.371

 

Exponential

0.663647

177.66

179.66

179.73

181.80

0.402

Data 2

Akash
Lindley

0.043876
0.028859

950.97 983.11

952.97
985.11

953.01
985.15

955.58
987.71

0.184
0.242

 

Exponential

0.014635

1044.87

1046.87

1046.91

1049.48

0.357

Data 3

Akash
Lindley

0.04151
0.027321

227.06
231.47

229.06
233.47

229.25
233.66

230.20
234.61

0.107
0.149

 

Exponential

0.013845

242.87

244.87

245.06

246.01

0.263

Data 4

Akash
Lindley

0.013514
0.00897

1255.83
1251.34

1257.83
1253.34

1257.87
1253.38

1260.43
1255.95

0.071
0.098

 

Exponential

0.004505

1280.52

1282.52

1282.56

1285.12

0.190

Data 5

Akash
Lindley

0.030045
0.019841

794.70
789.04

796.70
791.04

796.76
791.10

798.98
793.32

0.184
0.133

 

Exponential

0.010018

806.88

808.88

808.94

811.16

0.198

Data 6

Akash
Lindley

0.11961
0.077247

981.28
1041.64

983.28
1043.64

983.31
1043.68

986.18
1046.54

0.393
0.448

 

Exponential

0.04006

1130.26

1132.26

1132.29

1135.16

0.525

Data 7

Akash
Lindley

0.013263
0.008804

803.96
763.75

805.96
765.75

806.02
765.82

810.01
767.81

0.298
0.245

 

Exponential

0.004421

744.87

746.87

746.94

748.93

0.166

Data 8

Akash
Lindley

0.013423
0.00891

609.93
579.16

611.93
581.16

612.02
581.26

613.71
582.95

0.280
0.219

 

Exponential

0.004475

564.02

566.02

566.11

567.80

0.145

Data 9

Akash
Lindley

0.3105
0.196045

887.89
839.06

889.89
841.06

889.92
841.09

892.74
843.91

0.198
0.116

 

Exponential

0.106773

828.68

830.68

830.72

833.54

0.077

Data 10

Akash
Lindley

0.050293
0.033021

354.88
323.27

356.88
325.27

357.02
325.42

358.28
326.67

0.421
0.345

 

Exponential

0.016779

305.26

307.26

307.40

308.66

0.213

Data 11

Akash
Lindley

1.165719
0.823821

115.15
112.61

117.15
114.61

117.28
114.73

118.68
116.13

0.156
0.133

 

Exponential

0.532081

110.91

112.91

113.03

114.43

0.089

Data 12

Akash
Lindley

0.295277
0.186571

641.93
638.07

643.93
640.07

643.95
640.12

646.51
642.68

0.100
0.058

 

Exponential

0.101245

658.04

660.04

660.08

662.65

0.163

Data 13

Akash
Lindley

0.024734
0.01636

194.30
181.34

196.30
183.34

196.61
183.65

197.01
184.05

0.456
0.386

 

Exponential

0.008246

173.94

175.94

176.25

176.65

0.277

Data 14

Akash
Lindley

1.156923
0.816118

59.52
60.50

61.52
62.50

61.74
62.72

62.51
63.49

0.320
0.341

 

Exponential

0.526316

65.67

67.67

67.90

68.67

0.389

Data 15

Akash
Lindley
Exponential

0.097062
0.062988
0.032455

240.68
253.99
274.53

242.68
255.99
276.53

242.82
256.13
276.67

244.11
257.42
277.96

0.266
0.333
0.426

Data 16

Akash
Lindley
Exponential

0.964726
0.659000
0.407941

224.28
238.38
261.74

226.28
240.38
263.74

226.34
240.44
263.80

228.51
242.61
265.97

0.348
0.390
0.434

Table 3: MLE’s, -2ln L, AIC, AICC, BIC, K-S Statistics of the fitted distributions of data sets 1-16.

AIC=2lnL+2k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaamysaiaadoeacqGH9aqpcqGHsislcaaIYaGaciiBaiaac6gacaWG mbGaey4kaSIaaGOmaiaadUgaaaa@40D3@ , AICC=AIC+ 2k( k+1 ) ( nk1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGbb GaamysaiaadoeacaWGdbGaeyypa0JaamyqaiaadMeacaWGdbGaey4k aSscfa4aaSaaaOqaaKqzGeGaaGOmaiaadUgajuaGdaqadaGcbaqcLb sacaWGRbGaey4kaSIaaGymaaGccaGLOaGaayzkaaaabaqcfa4aaeWa aOqaaKqzGeGaamOBaiabgkHiTiaadUgacqGHsislcaaIXaaakiaawI cacaGLPaaaaaaaaa@4D49@ , BIC=2lnL+klnn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb GaamysaiaadoeacqGH9aqpcqGHsislcaaIYaGaciiBaiaac6gacaWG mbGaey4kaSIaam4AaiGacYgacaGGUbGaamOBaaaa@42EF@ and D= Sup x | F n ( x ) F 0 ( x ) | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGeb Gaeyypa0tcfa4aaCbeaOqaaKqzGeGaae4uaiaabwhacaqGWbaaleaa jugWaiaadIhaaSqabaqcfa4aaqWaaOqaaKqzGeGaamOraKqbaoaaBa aaleaajugWaiaad6gaaSqabaqcfa4aaeWaaOqaaKqzGeGaamiEaaGc caGLOaGaayzkaaqcLbsacqGHsislcaWGgbqcfa4aaSbaaSqaaKqzad GaaGimaaWcbeaajuaGdaqadaGcbaqcLbsacaWG4baakiaawIcacaGL PaaaaiaawEa7caGLiWoaaaa@5307@ , where k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGRb aaaa@3775@ the number of parameters, n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGUb aaaa@3778@ is the sample size and F n ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOram aaBaaajuaibaGaamOBaaqcfayabaWaaeWaaeaacaWG4baacaGLOaGa ayzkaaaaaa@3BA5@ is the empirical distribution function. The best distribution corresponds to lower values of 2lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqGHsi slcaaIYaGaciiBaiaac6gacaWGmbaaaa@3AE3@ , AIC, AICC, BIC, and K-S statistics. The fittings of Akash, Lindley and exponential distributions are based on maximum likelihood estimates (MLE).

Let t 1 , t 2 ,...., t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaacYcacaWG0bqc fa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiaacYcacaGGUaGaai Olaiaac6cacaGGUaGaaiilaiaadshajuaGdaWgaaWcbaqcLbmacaWG Ubaaleqaaaaa@47A9@ be a random sample of size n from exponential distribution. The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb aaaa@3756@ and the log likelihood function, lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaiaadYeaaaa@393A@ of exponential distribution are given by L= θ n e nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0JaeqiUdexcfa4aaWbaaSqabeaajugWaiaad6gaaaqcLbsa caWGLbqcfa4aaWbaaSqabeaajugWaiabgkHiTiaad6gacaaMc8Uaeq iUdeNaaGPaVlqadshagaqeaaaaaaa@480D@ and lnL=nlnθnθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaiaadYeacqGH9aqpcaWGUbGaciiBaiaac6gacqaH4oqCcqGH sislcaWGUbGaaGPaVlabeI7aXjaaykW7ceWG0bGbaebaaaa@468A@ . The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@ of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ of exponential distribution is the solution of the equation dlnL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGUbGaamitaaGcbaqcLbsacaWGKbGa eqiUdehaaiabg2da9iaaicdaaaa@3FC3@ and is given by θ ^ = 1 t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaiabg2da9KqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa bmiDayaaraaaaaaa@3CED@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG0b Gbaebaaaa@3796@ is the sample mean.

Let t 1 , t 2 ,...., t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaacYcacaWG0bqc fa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiaacYcacaGGUaGaai Olaiaac6cacaGGUaGaaiilaiaadshajuaGdaWgaaWcbaqcLbmacaWG Ubaaleqaaaaa@47A9@ be a random sample of size n from Lindley distribution. The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb aaaa@3756@ and the log likelihood function, lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaiaadYeaaaa@393A@ of Lindley distribution are given by L= ( θ 2 θ+1 ) n i=1 n ( 1+ t i ) e nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamitai abg2da9maabmaabaWaaSaaaeaacqaH4oqCdaahaaqabKqbGeaacaaI YaaaaaqcfayaaiabeI7aXjabgUcaRiaaigdaaaaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaamOBaaaajuaGdaqeWbqaamaabmaabaGaaGym aiabgUcaRiaadshadaWgaaqcfasaaiaadMgaaKqbagqaaaGaayjkai aawMcaaaqcfasaaiaadMgacqGH9aqpcaaIXaaabaGaamOBaaqcfaOa ey4dIunacaaMc8UaamyzamaaCaaabeqcfasaaiabgkHiTiaad6gaca aMc8UaeqiUdeNaaGPaVlqadshagaqeaaaaaaa@594B@ and lnL=nln( θ 2 θ+1 )+ i=1 n ln( 1+ t i ) nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaciiBai aac6gacaWGmbGaeyypa0JaamOBaiGacYgacaGGUbWaaeWaaeaadaWc aaqaaiabeI7aXnaaCaaabeqcfasaaiaaikdaaaaajuaGbaGaeqiUde Naey4kaSIaaGymaaaaaiaawIcacaGLPaaacqGHRaWkdaaeWbqaaiGa cYgacaGGUbWaaeWaaeaacaaIXaGaey4kaSIaamiDamaaBaaajuaiba GaamyAaaqcfayabaaacaGLOaGaayzkaaaajuaibaGaamyAaiabg2da 9iaaigdaaeaacaWGUbaajuaGcqGHris5aiabgkHiTiaad6gacaaMc8 UaeqiUdeNaaGPaVlqadshagaqeaaaa@5C47@ . The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@ of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ of Lindley distribution is the solution of the equation dlnL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGUbGaamitaaGcbaqcLbsacaWGKbGa eqiUdehaaiabg2da9iaaicdaaaa@3FC3@ and is given by θ ^ = ( t ¯ 1 )+ ( t ¯ 1 ) 2 +8 t ¯ 2 t ¯ ; t ¯ >0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaiabg2da9KqbaoaalaaakeaajugibiabgkHiTKqbaoaabmaa keaajugibiqadshagaqeaiabgkHiTiaaigdaaOGaayjkaiaawMcaaK qzGeGaey4kaSscfa4aaOaaaOqaaKqbaoaabmaakeaajugibiqadsha gaqeaiabgkHiTiaaigdaaOGaayjkaiaawMcaaKqbaoaaCaaaleqaba qcLbmacaaIYaaaaKqzGeGaey4kaSIaaGioaiaaykW7ceWG0bGbaeba aSqabaaakeaajugibiaaikdacaaMc8UabmiDayaaraaaaiaacUdaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ceWG0bGbaebacqGH +aGpcaaIWaaaaa@60B8@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG0b Gbaebaaaa@3796@ is the sample mean.

Let t 1 , t 2 ,...., t n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWG0b qcfa4aaSbaaSqaaKqzadGaaGymaaWcbeaajugibiaacYcacaWG0bqc fa4aaSbaaSqaaKqzadGaaGOmaaWcbeaajugibiaacYcacaGGUaGaai Olaiaac6cacaGGUaGaaiilaiaadshajuaGdaWgaaWcbaqcLbmacaWG Ubaaleqaaaaa@47A9@ be a random sample of size n from Akash distribution. The likelihood function, L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb aaaa@3756@ and the log likelihood function, lnL MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaiaadYeaaaa@393A@ of Akash distribution are given by L= ( θ 3 θ 2 +2 ) n i=1 n ( 1+ t i 2 ) e nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb Gaeyypa0tcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiabeI7aXLqb aoaaCaaaleqabaqcLbmacaaIZaaaaaGcbaqcLbsacqaH4oqCjuaGda ahaaWcbeqaaKqzadGaaGOmaaaajugibiabgUcaRiaaikdaaaaakiaa wIcacaGLPaaajuaGdaahaaWcbeqaaKqzadGaamOBaaaajuaGdaqeWb Gcbaqcfa4aaeWaaOqaaKqzGeGaaGymaiabgUcaRiaadshajuaGdaWg aaWcbaqcLbmacaWGPbaaleqaaKqbaoaaCaaaleqabaqcLbmacaaIYa aaaaGccaGLOaGaayzkaaaaleaajugWaiaadMgacqGH9aqpcaaIXaaa leaajugWaiaad6gaaKqzGeGaey4dIunacaaMc8UaamyzaKqbaoaaCa aaleqabaqcLbmacqGHsislcaWGUbGaaGPaVlabeI7aXjaaykW7ceWG 0bGbaebaaaaaaa@6A5C@ and lnL=nln( θ 3 θ 2 +2 )+ i=1 n ln( 1+ t i 2 ) nθ t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaciGGSb GaaiOBaiaadYeacqGH9aqpcaWGUbGaciiBaiaac6gajuaGdaqadaGc baqcfa4aaSaaaOqaaKqzGeGaeqiUdexcfa4aaWbaaSqabeaajugWai aaiodaaaaakeaajugibiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaI YaaaaKqzGeGaey4kaSIaaGOmaaaaaOGaayjkaiaawMcaaKqzGeGaey 4kaSscfa4aaabCaOqaaKqzGeGaciiBaiaac6gajuaGdaqadaGcbaqc LbsacaaIXaGaey4kaSIaamiDaKqbaoaaBaaaleaajugWaiaadMgaaS qabaqcfa4aaWbaaSqabeaajugWaiaaikdaaaaakiaawIcacaGLPaaa aSqaaKqzadGaamyAaiabg2da9iaaigdaaSqaaKqzadGaamOBaaqcLb sacqGHris5aiabgkHiTiaad6gacaaMc8UaeqiUdeNaaGPaVlqadsha gaqeaaaa@6BD2@ . The MLE θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacuaH4o qCgaqcaaaa@384B@ of the parameter θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaH4o qCaaa@383B@ of Akash distribution is the solution of the equation dlnL dθ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaaO qaaKqzGeGaamizaiGacYgacaGGUbGaamitaaGcbaqcLbsacaWGKbGa eqiUdehaaiabg2da9iaaicdaaaa@3FC3@ is the solution of following non-linear equation t ¯ θ 3 θ 2 +2 t ¯ θ6=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG0b GbaebacqaH4oqCjuaGdaahaaWcbeqaaKqzadGaaG4maaaajugibiab gkHiTiabeI7aXLqbaoaaCaaaleqabaqcLbmacaaIYaaaaKqzGeGaey 4kaSIaaGOmaiqadshagaqeaiabeI7aXjabgkHiTiaaiAdacqGH9aqp caaIWaaaaa@4A2A@ , where t ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsaceWG0b Gbaebaaaa@3796@ is the sample mean.

It is obvious from the goodness of fit of Akash, Lindley and exponential distributions that the Akash distribution provides better fit than the Lindley and exponential distributions in data-sets 2, 3, 6, 14, 15, and 16; the Lindley distribution gives better fit than the exponential and Akash distributions in data-sets 1, 4, 5 and 12; the exponential distribution gives better fit than the Lindley and the Akash distributions in data sets 7, 8, 9, 10, 11, and 13.

0.55

0.93

1.25

1.36

1.49

1.52

1.58

1.61

1.64

1.68

1.73

1.81

2

0.74

1.04

1.27

1.39

1.49

1.53

1.59

1.61

1.66

1.68

1.76

1.82

2.01

0.77

1.11

1.28

1.42

1.5

1.54

1.6

1.62

1.66

1.69

1.76

1.84

2.24

0.81

1.13

1.29

1.48

1.5

1.55

1.61

1.62

1.66

1.7

1.77

1.84

0.84

1.24

1.3

1.48

1.51

1.55

1.61

1.63

1.67

1.7

1.78

1.89

Data Set 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 [28].

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 [29] on the fatigue life of 6061 – T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data ( × MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey41aqlaaa@380D@ 10 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaaIXa GaaGimaKqbaoaaCaaaleqabaqcLbmacqGHsislcaaIZaaaaaaa@3B8D@ ) are presented below (after subtracting 65).

17.88

28.92

33

41.5

42.12

45.6

48.8

51.8

52

54.12

55.56

67.8

68.44

68.64

68.9

84.1

93.12

98.6

105

106

128

128

173.4

Data Set 3: The data set is from Lawless [30]. The data given arose in tests on endurance of deep groove ball bearings. The data are the number of million revolutions before failure for each of the 23 ball bearings in the life tests and they are.

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 [31] and arose in test on the cycle at which the Yarn failed. The data are the number of cycles until failure of the yarn and they are.

12

15

22

24

24

32

32

33

34

38

38

43

44

48

52

53

54

54

55

56

57

58

58

59

60

60

60

60

61

62

63

65

65

67

68

70

70

72

73

75

76

76

81

83

84

85

87

91

95

96

98

99

109

110

121

127

129

131

143

146

146

175

175

211

233

258

258

263

297

341

341

376

Data Set 5: This data represents the survival times (in days) of 72 guinna pigs infected with virulent tubercle bacilli, observed and reported by Bjerkedal [32].

19(16)

20(15)

21(14)

22(9)

23(12)

24(10)

25(6)    

26(9)

27(8)

28(5)

29(6)

30(4)

 31(3)

32(4)

33

34

35(4)

36(2)

37(2)

39

42   44

Data Set 6: This data is related with behavioral sciences, collected by Balakrishnan N, Victor Leiva & Antonio Sanhueza [33]: The scale “General Rating of Affective Symptoms for Preschoolers (GRASP)” measures behavioral and emotional problems of children, which can be classified with depressive condition or not according to this scale. A study conducted by the authors in a city located at      the south part of Chile has allowed collecting real data corresponding to the scores of the GRASP scale of children with frequency in parenthesis, which are.

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

12.2

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 [34] 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

0.2

2.23

3.52

4.98

6.97

9.02

13.29

0.4

2.26

3.57

5.06

7.09

9.22

13.8

25.74

0.5

2.46

3.64

5.09

7.26

9.47

14.24

25.82

0.51

2.54

3.7

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

0.9

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

1.4

3.02

4.34

5.71

7.93

11.79

18.1

1.46

4.4

5.85

8.26

11.98

19.13

1.76

3.25

4.5

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 [35].

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 [36], 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. [37], is vinyl chloride data obtained from clean up gradient monitoring wells in mg/l.

0.8,

0.8,

1.3,

1.5,

1.8,

1.9,

1.9,

2.1,

2.6,

2.7,

2.9,

3.1,

3.2,

3.3,

3.5,

3.6,

4.0,

4.1,

4.2,

4.2,

4.3,

4.3,

4.4,

4.4,

4.6,

4.7,

4.7,

4.8,

4.9,

4.9,

5.0,

5.3,

5.5,

5.7,

5.7,

6.1,

6.2,

6.2,

6.2,

6.3,

6.7,

6.9,

7.1,

7.1,

7.1,

7.1,

7.4,

7.6,

7.7,

8.0,

8.2,

8.6,

8.6,

8.6,

8.8,

8.8,

8.9,

8.9,

9.5,

9.6,

9.7,

9.8,

10.7,

10.9,

11.0,

11.0,

11.1,

11.2,

11.2,

11.5,

11.9,

12.4,

12.5,

12.9,

13.0,

13.1,

13.3,

13.6,

13.7,

13.9,

14.1,

15.4,

15.4,

17.3,

17.3,

18.1,

18.2,

18.4,

18.9,

19.0,

19.9,

20.6,

21.3,

21.4,

21.9,

23.0,

27.0,

31.6,

33.1,

38.5

Data Set 12: This data set represents the waiting times (in minutes) before service of 100 Bank customers and examined and analyzed by Ghitany et al. [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 [38].

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 [39].

18.83

20.8

21.66

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

44.05

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 [40].

1.312

1.314

1.479

1.552

1.7

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

2.14

2.179

2.224

2.24

2.253

2.27

2.272

2.274

2.301

2.301

2.359

2.382

2.382

2.426

2.434

2.435

2.478

2.49

2.511

2.514

2.535

2.554

2.566

2.57

2.586

2.629

2.633

2.642

2.648

2.684

2.697

2.726

2.77

2.773

2.8

2.809

2.818

2.821

2.848

2.88

2.954

3.012

3.067

3.084

3.09

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 [41,42,43,44].

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