The modeling and statistical analysis of lifetime data are crucial for statisticians and research workers in almost all applied sciences including engineering, medical sciences/biological sciences, insurance, finance, amongst others. One parameter lifetime distributions that are popular in Statistics literature for modeling lifetime data are exponential and Lindley distributions. An extensive study has been carried out by Shanker et al.¹ for modeling lifetime data using Lindley and exponential distributions and observed that there are many lifetime data where these distributions are not suitable from theoretical and applied point of view. Recently Shanker^2,3 has introduced one parameter Lifetime distributions namely “Aradhana distribution” and “Sujatha distribution” for modeling lifetime data.

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

Keywords: aradhana distribution, sujatha distribution, lindley distribution, exponential distribution, statistical properties, estimation of parameter, goodness of fit

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

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

Shanker² introduced a new one parameter continuous distribution named, ‘Aradhana distribution’ for modeling lifetime data from engineering and medical science. This distribution is a three- component mixture of an exponential $\left(\theta \right)$  distribution, a gamma $\left(2,\theta \right)$ distribution and a gamma $\left(3,\theta \right)$ distribution with their mixing proportions $\frac{{\theta }^2}{{\theta }^2+2\theta +2}$  , $\frac{2\theta }{{\theta }^2+2\theta +2}$  and $\frac{2}{{\theta }^2+2\theta +2}$  respectively. It has been shown by Shanker² that Aradhana distribution is flexible than the Lindley distribution for modeling lifetime data in reliability and in terms of its hazard rate shapes and it gives better fit than Akash, Shanker, Lindley and exponential distributions in modeling lifetime data. Shanker² has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability, amongst others. Shanker⁴ has also obtained a Poisson mixture of Aradhana distribution named, “Poisson-Aradhana distribution (PAD)”for modeling count data.

Shanker³ introduced another one parameter continuous distribution named, ‘Sujatha distribution’ for modeling lifetime data from engineering and medical science. This distribution is also a three-component mixture of an exponential $\left(\theta \right)$  distribution, a gamma $\left(2,\theta \right)$ distribution and a gamma $\left(3,\theta \right)$ distribution with their mixing proportions $\frac{{\theta }^2}{{\theta }^2+\theta +2}$ , $\frac{\theta }{{\theta }^2+\theta +2}$ and $\frac{2}{{\theta }^2+\theta +2}$ respectively. It has been shown by Shanker³ that Sujatha distribution is flexible than the Lindley distribution for modeling lifetime data in reliability and in terms of its hazard rate shapes and it gives better fit than Lindley and exponential distributions in modeling lifetime data. Shanker³ has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability, amongst others. Shanker⁵ has also obtained a Poisson mixture of Sujatha distribution named, “Poisson-Sujatha distribution (PSD)”for modeling count data.

The Lindley distribution is a two-component mixture of an exponential $\left(\theta \right)$ distribution and a gamma $\left(2,\theta \right)$  distribution with their mixing proportions $\frac{\theta }{\theta +1}$  and $\frac{1}{\theta +1}$ respectively 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, mixed, modified and its detailed applications in reliability and other fields of knowledge by different researchers including Sankaran,⁸ Zakerzadeh & Dolati,⁹ Nadarajah et al.,¹⁰ Deniz & Ojeda,¹¹ Bakouch et al.,¹² Shanker & Mishra,^13,14,15 Shanker & Amanuel,¹⁶ Ghitany et al.,¹⁷ Shanker et al.,^1,18-21 are some among others.

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

Let $T$ 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, $F\left(t\right)$ , hazard rate function, $h\left(t\right)$ , mean residual life function, $m\left(t\right)$ , mean ${\mu }_1{}^{\prime }$ , variance ${\mu }_2$ , 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 Aradhana and Sujatha distributions are summarized in Table 1 and of Lindley and exponential distributions in Table 2.

A table of values for coefficient of variation (C.V.), coefficient of Skewness $\left(\sqrt{{\beta }_1}\right)$ , coefficient of Kurtosis $\left({\beta }_2\right)$ , and index of dispersion $\left(\gamma \right)$  for Aradhana, Sujatha and Lindley distributions for various values of their parameter for comparative study are summarized in the Table 3.

The condition under which Aradhana, Sujatha, Lindley and exponential distributions are Over-dispersion $\left(\mu <{\sigma }^2\right)$ , equi-dispersion $\left(\mu ={\sigma }^2\right)$  and under-dispersion $\left(\mu >{\sigma }^2\right)$  of Aradhana, Sujatha, Lindley and exponential distributions for varying values of their parameter $\theta$ are presented in Table 4.

Graphs of coefficient of variation (C.V), coefficient of skewness ( $\sqrt{{\beta }_1}$ ) coefficient of kurtosis ( ${\beta }_2$ ) and index of dispersion ( $\gamma$ ) for Aradhana, Sujatha, and Lindley distributions are presented for varying values of their parameter $\theta$  in Figure 1.

Estimate of the parameter of Aradhana distribution

Let $\left(t_1,t_2,t_3,\mathrm{...},t_n\right)$  be a random sample from Aradhana distribution. The maximum likelihood estimate (MLE) $\widehat{\theta }$  of $\theta$  and the method of moment estimate (MOME) $\tilde{\theta }$ of $\theta$  is the solution of the following cubic equation

$\overline{t}{\theta }^3+\left(2\overline{t}-1\right){\theta }^2+2\left(\overline{t}-2\right)\theta -6=0$ .

Estimate of the parameter of Sujatha distribution

Let $\left(t_1,t_2,t_3,\mathrm{...},t_n\right)$  be random sample from Sujatha distribution. The maximum likelihood estimate (MLE) $\widehat{\theta }$  of $\theta$  and the method of moment estimate (MOME) $\tilde{\theta }$ of $\theta$  is the solution of the following cubic equation

  $\overline{t}{\theta }^3+\left(\overline{t}-1\right){\theta }^2+2\left(\overline{t}-1\right)\theta -6=0$

Estimate of the parameter of Lindley distribution

Let $\left(t_1,t_2,\mathrm{....},t_n\right)$  be a random sample of size $n$  from Lindley distribution. The MLE $\widehat{\theta }$  of $\theta$  and MOME $\tilde{\theta }$ of $\theta$  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.

Estimate of the parameter of Exponential distribution

Let $\left(t_1,t_2,\mathrm{....},t_n\right)$  be a random sample of size $n$  from exponential distribution. The MLE $\widehat{\theta }$  of $\theta$  and MOME $\tilde{\theta }$ of $\theta$  is given by $\widehat{\theta }=\frac{1}{\overline{t}}$ , where $\overline{t}$ is the sample mean.

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

In order to compare Aradhana, Sujatha, Lindley and exponential distributions, $-2\ln L$ , 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 and presented in Table 5. 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 $k$ = number of parameters, $n$ = the sample size and $F_n\left(x\right)$ is the empirical distribution function. The best distribution is the distribution corresponding to lower values of $-2\ln L$ , AIC, AICC, BIC, and K-S statistics.

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

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

Figure 1 Graphs of coefficient of variation (C.V), coefficient of skewness ( $\sqrt{{\beta }_1}$ ) coefficient of kurtosis ( ${\beta }_2$ ) and index of dispersion ( $\gamma$ ) for Aradhana, Sujatha, and Lindley distributions are for varying values of their parameter $\theta$ .

None.

Authors declare that there are no conflicts of interests.

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