A Generalized Akash Distribution

various mathematical and statistical properties and showed that in many ways (1.1) provides a better model for modeling lifetime data from medical science and engineering than Lindley [2] and exponential distributions. Shanker et al. [3,4] have detailed comparative and critical study on Akash, Lindley and exponential distributions for modeling lifetime data from biomedical science and engineering and observed that Akash distribution provides better fit than both Lindley and exponential distributions in almost all lifetime datasets.


Introduction
Shanker [1] introduced a one-parameter lifetime distribution, known as Akash distribution, defined by its probability density function (pdf) and cumulative distribution function (cdf)

Statistical Constants
The moment generating function (mgf) about origin of GAD (2.1) can be obtained as It can be easily verified that these statistical constants of GAD reduces to the corresponding statistical constants of Akash and exponential distributions atr 1 α = and 0 α = respectively. Graphs of coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of GAD for varying values of parameters θ and α have been drawn and presented in figure 3.

Stochastic Ordering
Stochastic ordering of positive continuous random variables is an important tool for judging the comparative behaviour. A random variable X is said to be smaller than a random variable Y in the The following results due to Shaked and Shanthikumar [8] are well known for establishing stochastic ordering of distributions The GAD is ordered with respect to the strongest 'likelihood ratio' ordering as established in the following theorem: This theorem shows the flexibility of GAD over Akash and exponential distributions.

Mean Deviations
The amount of scatter in a population is generally measured to some extent by the totality of deviations usually from the mean and the median. These are known as the mean deviation about the mean and the mean deviation about the median defined by can be calculated using the following simplified relationships Using pdf (2.1) and expression for the mean of GAD, we get

Distribution of Order Statistics
Let 1 2 , ,..., n X X X be a random sample of size n from GAD denote the corresponding order statistics. The pdf and the cdf of the k th order statistic, say respectively, for 1, 2, 3, ..., k n = .
Thus, the pdf and the cdf of k th order statistics of GAD are obtained as

Renyi Entropy Measure
Entropy of a random variable X is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy [9]. If X is a continuous random variable having pdf ( ) . f , then Renyi entropy is defined as Thus, the Renyi entropy of GAD (2.1) can be obtained as

Bonferroni and Lorenz curves
The Bonferroni and Lorenz curves [10] 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

Hazard Rate Function and Mean Residual Life Function
For a continuous distribution with pdf ( ) f x and cdf ( )

Stress-Strength Reliability
The stress-strength reliability describes the life of a component which has random strength X that is subjected to a random stress X . When the stress applied to it exceeds the strength, the component fails instantly and the component will function satisfactorily till X Y > . Therefore, is a measure of the component reliability and known as stress-strength parameter in statistical literature. It has wide applications in almost all areas of knowledge especially in medical science and engineering.
Let X and X be independent strength and stress random variables having GAD (2.1) with parameter ( )  ( It can be easily verified that at ( )

Estimates from Moments
Since the GAD (2.1) has two parameters to be estimated, the first two moments about origin are required to estimate its parameters. Using the first two moments about origin of GAD
The likelihood function, L of (2.1) is given by ( ) , where x is the sample mean.
These two natural log likelihood equations do not seem to be solved directly because these equations cannot be expressed in closed forms. However, the Fisher's scoring method can be applied to solve these equations. We have where 0 θ and 0 α are the initial values of θ and α , respectively.
These equations are solved iteratively till sufficiently close values of θ and α are obtained. The initial values of the parameters θ and α are taken from MOME estimates.

An Illustrative Example
The following data set represents the failure times (in minutes) for a sample of 15 electronic components in an accelerated life test, Lawless [11]