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. The classical lifetime distributions namely exponential and Lindley (1958) distributions are popular in statistics for modeling lifetime data. But these two classical lifetime distributions are not suitable from theoretical and applied point of view. Shanker et al (2015) have done a critical and comparative study regarding the modeling of lifetime data using both exponential and Lindley distributions and found that there are several lifetime data where these classical lifetime distributions are not suitable due to their shapes, hazard rate functions and mean residual life functions, amongst others. Recently, a number of one parameter lifetime distributions have been introduced by Shanker (2015 a, 2015 b, 2016 a, 2016 b, 2016 c, 2016 d, 2016 e) namely Akash, Shanker, Amarendra, Aradhana, Sujatha, Devya and Shambhu. Although these lifetime distributions give better fit than the classical exponential and Lindley distributions, there are still some lifetime data where these distributions are not suitable due to their theoretical and applied point of view.

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

(1.1)

We would call this distribution, “Akshaya distribution”. The corresponding cumulative distribution function (c.d.f.) of (1.1) can be obtained as

(1.2)

The graph of the p.d.f. and the c.d.f. of Akshaya distribution for different values of the parameter are shown in figures 1 and 2.

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

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

2. Moments and Related Measures

The moment generating function of Akshaya distribution (1.1) can be obtained as

Thus the th moment about origin, as given by the coefficient of in of Akshaya distributon (1.1) has been obtained as

The first four moments about origin of Akshaya distribution (1.1) are obtained as

,

Thus the moments about mean of the Akshaya distribution (1.1) are obtained as

The coefficient of variation coefficient of skewness coefficient of kurtosis and index of dispersion of Akshaya distribution (1.1) are thus obtained as

The condition under which Akshaya distribution is over-dispersed, equi-dispersed, and under-dispersed has been studied along with condition under which Akash, Shanker, Amarendra, Aradhana, Sujatha, Devya, Shambhu, Lindley and exponential distributions are over-dispersed, equi-dispersed, and under-dispersed and are presented in table 1.

Table 1. Over-dispersion, equi-dispersion and under-dispersion of Akshaya, Akash, Shanker, Amarendra, Aradhana, Sujatha, Devya, Shambhu, Lindley and exponential distributions for parameter θ

3. Hazard Rate Function and Mean Residual Life Function

Let be a continuous random variable with p.d.f. and c.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 Akshaya distribution (1.1) are obtained as

(3.3)

and

(3.4)

It can be easily seen that and It is also obvious from the graphs of and that is an increasing function of and whereas is firstly increasing and then decreasing function of and

The graphs of the hazard rate function and mean residual life function of Akshaya distribution (1.1) are shown in figures 3 and 4.

Figure 3. Graph of hazard rate function of Akshaya distribution for varying values of parameter θ

Figure 4. Graph of mean residual life function of Akshaya distribution for varying values of parameter θ

4. Stochastic Orderings

Stochastic ordering of positive continuous random variable 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 if for all

(ii) hazard rate order if for all

(iii) mean residual life order if for all

(iv) likelihood ratio order if decreases in

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

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

Theorem: Let Akshaya distribution and Akshaya distribution . If then and hence and

Proof: We have

Now

This gives

Thus for This means that and hence and

5. Mean Deviations about Mean and Median

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 and can be computed using the following simplified relationships

(5.1)

and

(5.2)

Using p.d.f. (1.1) and expression for the mean of Akshaya distribution, 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 Akshaya distribution (1.1), after a little algebraic simplification, are obtained as

(5.5)

(5.6)

6. 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

(6.1)

(6.2)

respectively or equivalently

(6.3)

(6.4)

respectively, where and

The Bonferroni and Gini indices are thus defined as

(6.5)

(6.6)

respectively.

Using p.d.f. of Akshaya distribution (1.1), we get

(6.7)

Now using equation (6.7) in (6.1) and (6.2), we get

(6.8)

(6.9)

Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini indices of Akshaya distribution (1.1) are obtained as

(6.10)

(6.11)

7. Estimation of Parameter

7.1. Maximum Likelihood Estimate (MLE)

Let be a random sample from Akshaya distribution (1.1). The likelihood function, of Akshaya distribution is given by

The natural log likelihood function is thus obtained as

Now

where is the sample mean.

The maximum likelihood estimate, of is the solution of the equation and so it can be obtained by solving the following fourth degree polynomial equation

(7.1.1)

7.2. Method of Moment Estimate (MOME)

Equating the population mean of the Akshaya distribution to the corresponding sample mean, the MOME of is same as given by equation (7.1.1)

8. Applications and Goodness of Fit

The Akshaya distribution has been fitted to a number of lifetime data set from biomedical science and engineering. In this section, we present the goodness of fit of Akshaya distribution to a real lifetime data and compare its goodness of fit with the one parameter Akash, Shanker, Amarendra, Aradhana, Sujatha, Devya, Shambhu, Lindley and exponential distributions.

The 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:

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, 3.0, 1.7, 2.3, 1.6, 2.0

In order to compare lifetime distributions, (Akaike Information Criterion) and K-S Statistics (Kolmogorov-Smirnov Statistics) for the above data set have been computed. The formulae for computing AIC and K-S Statistics are as follows:

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

The best distribution is the distribution which corresponds to lower values of and K-S statistics. The and standard error, of AIC and K-S Statistic of the fitted distributions are presented in the following table 2.

Table 2.       and K-S Statistics of the fitted distributions of the given data

It can be easily seen from above table that Akshaya distribution gives better fit than the fit given by Akash, Shanker, Amarendra, Aradhana, Sujatha, Devya, Shambhu, Lindley and exponential distributions and hence it can be considered as an important lifetime distribution over these lifetime distributions.

9. Conclusions

A new one parameter lifetime distribution named, “Akshaya distribution” has been suggested for modeling lifetime data 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, Bonferroni and Lorenz curves have been discussed. The method of moments and the method of maximum likelihood estimation have also been discussed for estimating its parameter. Finally, the goodness of fit using K-S Statistics (Kolmogorov-Smirnov Statistics) for a real lifetime data have been presented to demonstrate its applicability over Akash, Shanker, Amarendra, Aradhana, Sujatha, Devya, Shambhu, Lindley and exponential distributions.

ACKNOWLEDGEMENTS

The author is grateful to the editor- in - chief and the anonymous reviewer for helpful comments which improved the quality of the paper.