1. Introduction

The statistical analysis and modeling of lifetime data are essential in almost all applied sciences including, biomedical science, engineering, finance, and insurance, amongst others. A number of one parameter continuous distributions for modeling lifetime data has been introduced in statistical literature including Akash, Shanker, exponential, Lindley, gamma, lognormal, and Weibull. The Akash, Shanker, exponential, Lindley and the Weibull distributions are more popular 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 each of Akash, Shanker, Lindley and exponential distributions have one parameter, the Akash, Shanker, and Lindley distribution has one advantage over the exponential distribution that the exponential distribution has constant hazard rate whereas the Akash, Shanker, and Lindley distributions has monotonically increasing hazard rate. Further, it has been shown by Shanker (2015a, 2015 b) that Akash and Shanker distributions gives much closer fit in modeling lifetime data than Lindley and exponential distributions.

The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.) of Lindley (1958) distribution are given by

(1.1)

(1.2)

The density (1.1) is a two-component mixture of an exponential distribution having scale parameter and a gamma distribution having shape parameter 2 and scale parameter with their mixing proportions and respectively. 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 (2008). The Lindley distribution has been generalized, extended and modified along with their applications in modeling lifetime data from different fields of knowledge by different researchers including Zakerzadeh and Dolati (2009), Nadarajah et al (2011), Deniz and Ojeda (2011), Bakouch et al (2012), Shanker and Mishra (2013 a, 2013 b), Shanker and Amanuel (2013), Shanker et al (2013), Elbatal et al (2013), Ghitany et al (2013), Merovci (2013), Liyanage and Pararai (2014), Ashour and Eltehiwy (2014), Oluyede and Yang (2014), Singh et al (2014), Sharma et al (2015 a, 2015 b), Shanker et al (2015 a, 2015 b), Alkarni (2015), Pararai et al (2015), Abouammoh et al (2015) are some among others.

Shanker (2015 a) has introduced one parameter Akash distribution for modeling lifetime data defined by its p.d.f.

and c.d.f.

(1.3)

(1.4)

Shanker (2015 a) has shown that density (1.3) is a two-component mixture of an exponential distribution with scale parameter and a gamma distribution having shape parameter 3 and a scale parameter with their mixing proportions and respectively. Shanker (2015 a) has discussed its various mathematical and statistical properties including its shape, moment generating function, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic orderings, mean deviations, distribution of order statistics, Bonferroni and Lorenz curves, Renyi entropy measure, stress-strength reliability, amongst others. Shanker et al (2015 c) has detailed study about modeling of various lifetime data from different fields using Akash, Lindley and exponential distributions and concluded that Akash distribution has some advantage over Lindley and exponential distributions. Further, Shanker (2015 c) has obtained Poisson mixture of Akash distribution named Poisson-Akash distribution (PAD) and discussed its various mathematical and statistical properties, estimation of its parameter and applications for various count data-sets.

The probability density function and the cumulative distribution function of Shanker distribution introduced by Shanker (2015 b) are given by

(1.5)

(1.6)

Shanker (2015 b) has shown that density (1.5) is a two-component mixture of an exponential distribution with scale parameter and a gamma distribution having shape parameter 2 and a scale parameter with their mixing proportions respectively. Shanker (2015 b) 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 , some amongst others. Further, Shanker (2015 d) has obtained Poisson mixture of Shanker distribution named Poisson-Shanker distribution (PSD) and discussed its various mathematical and statistical properties, estimation of its parameter and applications for various count data-sets.

Although Akash, Shanker, Lindley and exponential distributions have been used to model various lifetime data from biomedical science and engineering, there are many situations where these distributions may not be suitable from theoretical or applied point of view.

In search for a new lifetime distribution, we have proposed a new lifetime distribution which is better than Akash, Shanker, Lindley and exponential distributions for modeling lifetime data by considering a three-component mixture of an exponential distribution having scale parameter , a gamma distribution having shape parameter 2 and scale parameter , and a gamma distribution with shape parameter 3 and scale parameter with their mixing proportions respectively. The probability density function (p.d.f.) of a new one parameter lifetime distribution can be introduced as

(1.7)

We would call this distribution, “Aradhana distribution”. The corresponding cumulative distribution function (c.d.f.) of (1.7) can be obtained as

(1.8)

The graphs of the p.d.f. and the c.d.f. of Aradhana distributions for different values of are shown in figures 1 and 2.

Figure 1. Graph of the pdf of Aradhana distribution for different values of parameter

Figure 2. Graph of the cdf of Aradhana distribution for different values of parameter

2. Moment Generating Function, Moments and Related Measures

The moment generating function of Aradhana distribution (1.7) can be obtained as

Thus the moment about origin, as given by the coefficient of of Aradhana distributon (1.7) has been obtained as

and so the first four moments about origin as

Thus the moments about mean of the Aradhana distribution (1.7) are obtained as

The coefficient of variation coefficient of skewness coefficient of kurtosis and Index of dispersion of Aradhana distribution (1.7) are thus obtained as

Table 1. Over-dispersion, equi-dispersion and under-dispersion of Aradhana, Akash, Shanker, Lindley, and exponential distributions for varying values of their parameter

3. Hazard Rate Function and Mean Residual Life Function

Let be a continuous random variable with p.d.f. The hazard rate function (also known as the failure rate function) and the mean residual life function of are respectively defined as

(3.1)

(3.2)

The corresponding hazard rate function, and the mean residual life function, of the Aradhana distribution (1.7) are obtained as

(3.3)

(3.4)

It can be easily seen that It is also obvious from the graphs of is an increasing function of is a decreasing function of

The graphs of the hazard rate function and mean residual life function of Aradhana distribution (1.7) are shown in figures 3 and 4.

Figure 3. Graph of hazard rate function of Aradhana distribution for different values of parameter

Figure 4. Graph of mean residual life function of Aradhana distribution for different values of parameter

4. Stochastic Orderings

Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable is said to be smaller than a random variable in the

(i) stochastic order for all

(ii) hazard rate order for all

(iii) mean residual life order if for all

(iv) likelihood ratio order decreases in

The following results due to Shaked and Shanthikumar (1994) are well known for establishing stochastic ordering of distributions

The Aradhana distribution is ordered with respect to the strongest ‘likelihood ratio’ ordering as shown in the following theorem:

Theorem: Let Aradhana distributon and Aradhana distribution . If then and hence and

Proof: We have

Now

This gives

Thus for This means that and hence

5. Mean Deviations

The amount of scatter in a population is measured to some extent by the totality of deviations usually from their mean and median. These are known as the mean deviation about the mean and the mean deviation about the median and are defined as

and respectively, where and The measures can be computed using the following simplified relationships

(5.1)

and

(5.2)

Using p.d.f. (1.7) and expression for the mean of Aradhana distribution (1.7), we get

(5.3)

(5.4)

Using expressions (5.1), (5.2), (5.3) and (5.4), the mean deviation about mean, and the mean deviation about median, of Aradhana distribution (1.7), after a little algebraic simplification, are obtained as

(5.5)

(5.6)

6. Order Statistics

Let be a random sample of size from Aradhana distribution (1.7). Let denote the corresponding order statistics. The p.d.f. and the c.d.f. of the order statistic, say are given by

and

,respectively, for

Thus, the p.d.f. and the c.d.f of order statistics of Aradhana distribution (1.7) are obtained as

and

7. Bonferroni and Lorenz Curves

The Bonferroni and Lorenz curves (Bonferroni, 1930) and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

(7.1)

(7.2)

respectively or equivalently

(7.3)

(7.4)

respectively, where

The Bonferroni and Gini indices are thus defined as

(7.5)

(7.6)

respectively.

Using p.d.f. (1.7), we get

(7.7)

Now using equation (7.7) in (7.1) and (7.2), we get

(7.8)

(7.9)

Now using equations (7.8) and (7.9) in (7.5) and (7.6), the Bonferroni and Gini indices of Aradhana distribution (1.7) are obtained as

(7.10)

(7.11)

8. Renyi Entropy

An entropy of a random variable is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy (1961). If is a continuous random variable having probability density function then Renyi entropy is defined as

where

Thus, the Renyi entropy for the Aradhana distribution (1.7) can be obtained as

9. Stress-strength Reliability

The stress - strength reliability describes the life of a component which has random strength that is subjected to a random stress . When the stress applied to it exceeds the strength , the component fails instantly and the component will function satisfactorily till . Therefore, is a measure of component reliability and is known as stress-strength reliability in statistical literature. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.

Let be independent strength and stress random variables having Aradhana distribution (1.7) with parameter respectively. Then the stress-strength reliability can be obtained as

.

10. Estimation of Parameter

10.1. Maximum Likelihood Estimation

Let be a random sample from Aradhana distribution (1.7). The likelihood function, of Aradhana distribution (1.7) is given by

The natural log likelihood function is thus obtained as

Now

where is the sample mean.

The maximum likelihood estimate, is the solution of the equation and so it can be obtained by solving the following non-linear equation

(10.1.1)

10.2. Method of Moment Estimation

Equating the population mean of the Aradhana distribution to the corresponding sample mean, the method of moment (MOM) estimate, is same as given by equation (10.1.1).

11. Applications and Goodness of Fit

The Aradhana distribution has been fitted to a number of lifetime data-sets relating to biomedical science and engineering. In this section, we present the fitting of Aradhana distribution to two real lifetime data- sets and compare its goodness of fit with the one parameter Akash, Shanker, Lindley and exponential distributions.

Data set 1: The first data - set represents the lifetime’s data relating to relief times (in minutes) of 20 patients receiving an analgesic and reported by Gross and Clark (1975, P. 105). The data are as follows:

Data set 2: The second data - set is the strength data of glass of the aircraft window reported by Fuller et al (1994)

In order to compare the goodness of fit of Aradhana, Akash, Shanker, Lindley and exponential distributions, AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected), BIC (Bayesian Information Criterion), K-S Statistics (Kolmogorov-Smirnov Statistics) for two real lifetime data - sets have been computed and presented in table 2. The formulae for computing AIC, AICC, BIC, and K-S Statistics are as follows:

and where the number of parameters, the sample size and is the empirical distribution function.

The best distribution for modeling lifetime data is the distribution corresponding to lower values of AIC, AICC, BIC, and K-S statistics

Table 2. MLE’s,       AIC, AICC, BIC, and K-S Statistics of the fitted distributions of data -sets 1 and 2

It can be easily seen from above table that in data-set 1, Aradhana distribution gives better fitting than Akash, Shanker, Lindley, and exponential distributions and in data-set 2, Akash distribution gives better fitting than Aradhana, Shanker, Lindley and exponential distributions.

12. Conclusions

A new one parameter lifetime distribution named, “Aradhana distribution” has been suggested for modeling lifetime data-sets from engineering and medical science. Its important mathematical and statistical properties including shape, moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, order statistics have been discussed. Further, expressions for Bonferroni and Lorenz curves, Renyi entropy measure and stress-strength reliability of the suggested distribution have been derived. The method of moments and the method of maximum likelihood estimation have also been discussed for estimating its parameter. Finally, the goodness of fit test using K-S Statistics (Kolmogorov-Smirnov Statistics) for two real lifetime data- sets have been presented to demonstrate the applicability and comparability of Aradhana, Akash, Shanker, Lindley and exponential distributions for modeling lifetime data - sets.

ACKNOWLEDGMENTS

The author expresses his gratefulness to the anonymous reviewer for constructive comments.