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

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 Shanker¹ 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 Shanker¹ has flexibility over both Lindley and exponential distributions.

1. Exponential distribution

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 Epstein⁴ 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 & Sobel³ and partly because it appeared suitable for representing the lifetimes of many things such as various types of manufactured items Davis.²

2. Lindley distribution

The Lindley distribution is a two-component mixture of an exponential distribution having scale parameter $\theta$  and a gamma distribution having shape parameter 2 and scale parameter $\theta$  with mixing proportions $\frac{\theta }{\theta +1}$ and $\frac{1}{\theta +1}$  and is given by Lindley⁶ 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 Hussain⁸ 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.

3. Akash distribution

Shanker¹ 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 $\theta$  and a gamma distribution having shape parameter 3 and scale parameter $\theta$  with mixing proportions $\frac{{\theta }^2}{{\theta }^2+2}$ and $\frac{1}{{\theta }^2+2}$ and has been shown by Shanker¹ that Akash distribution gives better fit than Lindley and exponential distributions in modeling lifetime data.

Let $T$ be a continuous random variable representing the lifetimes of individuals in some population. The expressions for probability density function, $f\left(t\right)$  , cumulative distribution function, $F\left(t\right)$  , survival function, $S\left(t\right)$ , hazard rate function, $h\left(t\right)$ , mean residual life function, $m\left(t\right)$ , mean ${\mu }_1{}^{\prime }$ , variance ${\mu }_2$ , third moment about mean ${\mu }_3$  , fourth moment about mean ${\mu }_4$ , coefficient of variation (C.V.), coefficient of Skewness $\left(\sqrt{{\beta }_1}\right)$ , coefficient of Kurtosis $\left({\beta }_2\right)$ , and index of dispersion $\left(\gamma \right)$  of exponential, Lindley and Akash distributions are summarized in the following .

It can be easily verified that the Akash distribution is over- dispersed $\left(\mu <{\sigma }^2\right)$ , equi-dispersed $\left(\mu ={\sigma }^2\right)$  and under-dispersed $\left(\mu >{\sigma }^2\right)$  for $\theta <(=)>{\theta }^{\ast }=1.515400063$  respectively. Further, Lindley distribution is over- dispersed $\left(\mu <{\sigma }^2\right)$ , equi-dispersed $\left(\mu ={\sigma }^2\right)$  and under-dispersed $\left(\mu >{\sigma }^2\right)$  for $\theta <(=)>{\theta }^{\ast }=1.170086487$  respectively, whereas as exponential distribution is over- dispersed $\left(\mu <{\sigma }^2\right)$ , equi-dispersed $\left(\mu ={\sigma }^2\right)$  and under- dispersed $\left(\mu >{\sigma }^2\right)$  for $\theta <(=)>{\theta }^{\ast }=1$  respectively.

A table of values for coefficient of variation (C.V.), coefficient of Skewness $\left(\sqrt{{\beta }_1}\right)$ , coefficient of Kurtosis $\left({\beta }_2\right)$ , and index of dispersion $\left(\gamma \right)$  for Akash and Lindley distributions for various values of their parameter for comparative study are summarized in the Table 2.

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, $-2\ln L$ , 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:

$AIC=-2\ln L+2k$ , $AICC=AIC+\frac{2k\left(k+1\right)}{\left(n-k-1\right)}$ , $BIC=-2\ln L+k\ln n$ and $D=\underset{x}{\text{Sup}}\left|F_n\left(x\right)-F_0\left(x\right)\right|$ , where $k$ the number of parameters, $n$ is the sample size and $F_n\left(x\right)$ is the empirical distribution function. The best distribution corresponds to lower values of $-2\ln L$ , AIC, AICC, BIC, and K-S statistics. The fittings of Akash, Lindley and exponential distributions are based on maximum likelihood estimates (MLE).

Let $t_1,t_2,\mathrm{....},t_n$ be a random sample of size n from exponential distribution. The likelihood function, $L$ and the log likelihood function, $\ln L$ of exponential distribution are given by $L={\theta }^n{e}^{-n\theta \overline{t}}$ and $\ln L=n\ln \theta -n\theta \overline{t}$ . The MLE $\widehat{\theta }$ of the parameter $\theta$ of exponential distribution is the solution of the equation $\frac{d\ln L}{d\theta }=0$ and is given by $\widehat{\theta }=\frac{1}{\overline{t}}$ , where $\overline{t}$ is the sample mean.

Let $t_1,t_2,\mathrm{....},t_n$ be a random sample of size n from Lindley distribution. The likelihood function, $L$ and the log likelihood function, $\ln L$ of Lindley distribution are given by $L={\left(\frac{{\theta }^2}{\theta +1}\right)}^n{\displaystyle \prod _{i=1}^n\left(1+t_i\right)}{e}^{-n\theta \overline{t}}$ and $\ln L=n\ln \left(\frac{{\theta }^2}{\theta +1}\right)+{\displaystyle \sum _{i=1}^n\ln \left(1+t_i\right)}-n\theta \overline{t}$ . The MLE $\widehat{\theta }$ of the parameter $\theta$ of Lindley distribution is the solution of the equation $\frac{d\ln L}{d\theta }=0$ and is given by $\widehat{\theta }=\frac{-\left(\overline{t}-1\right)+\sqrt{{\left(\overline{t}-1\right)}^2+8\overline{t}}}{2\overline{t}};\overline{t}>0$ , where $\overline{t}$ is the sample mean.

Let $t_1,t_2,\mathrm{....},t_n$ be a random sample of size n from Akash distribution. The likelihood function, $L$ and the log likelihood function, $\ln L$ of Akash distribution are given by $L={\left(\frac{{\theta }^3}{{\theta }^2+2}\right)}^n{\displaystyle \prod _{i=1}^n\left(1+t_i{}^2\right)}{e}^{-n\theta \overline{t}}$ and $\ln L=n\ln \left(\frac{{\theta }^3}{{\theta }^2+2}\right)+{\displaystyle \sum _{i=1}^n\ln \left(1+t_i{}^2\right)}-n\theta \overline{t}$ . The MLE $\widehat{\theta }$ of the parameter $\theta$ of Akash distribution is the solution of the equation $\frac{d\ln L}{d\theta }=0$ is the solution of following non-linear equation $\overline{t}{\theta }^3-{\theta }^2+2\overline{t}\theta -6=0$ , where $\overline{t}$ 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.

Acknowledgments

Conflicts of interest

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