On Quasi Lindley Distribution and Its Applications to Model Lifetime Data

In this paper mathematical and statistical properties including moment generating function, mean deviations about mean and median, order statistics, Bonferroni and Lorenz curves, Renyi entropy and stress strength reliability of quasi Lindley distribution (QLD) introduced by Shanker and Mishra (2013 a) have been derived and discussed. The goodness of fit of QLD over exponential and Lindley distributions have been illustrated with five real lifetime data-sets and found that QLD provides better fit than exponential and Lindley distributions.


Introduction
Lindley distribution, introduced in the context of Bayesian analysis as a counter example of fiducial statistics, having probability density function (p.d.f) and cumulative distribution function (c.d.f) (1.1) (1.2) has been introduced by Lindley (1958). A detailed study about its important mathematical and statistical 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, modified and mixed with other discrete distributions by different researchers including Zakerzadeh At α = θ , these moments reduce to the corresponding moments of Lindley distribution. Shanker and Mishra (2013 a) have derived and discussed some of its mathematical properties including its shape, moments, coefficient of variation, coefficient of skewness and kurtosis, hazard rate function, mean residual life function and stochastic orderings. They have also discussed the estimation of its parameters using maximum likelihood estimation and method of moments and its goodness of fit over Lindley and exponential distributions. It has been observed that many important mathematical and statistical properties of this distribution have not been derived and studied.
In the present paper some of the important mathematical and statistical properties including moment generating function, mean deviations about mean and median, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure and stress strength reliability of QLD of Shanker and Mishra (2013 a) have been derived and discussed. Its goodness of fit over exponential and Lindley distributions have been illustrated with some real lifetime data-sets and found that QLD gives better fit than exponential and Lindley distributions.

Moment Generating Function
The moment generating function,

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 and are known as the mean deviation about the mean and the mean deviation about the median, and are defined as 1 It can be easily verified that expressions (2.2.5) and (2.2.6) of QLD (1.3) reduce to the corresponding expressions of Lindley distribution at α = θ .

Distribution of Order Statistics
Let ( ) 1 2 n X , X ,..., X be a random sample of size n from

Bonferroni and Lorenz Curves and Indices
The Bonferroni and Lorenz curves (Bonferroni, 1930) and Bonferroni and Gini indices have much applications in economics to study income and poverty. But now a days these indices have many applications in other fields of knowledge including reliability, demography, insurance, medicine and engineering. The Bonferroni

Renyi Entropy Measure
The entropy of a random variable X is a measure of variation of uncertainty. A popular entropy measure is Renyi entropy (1961

Method of Moment Estimate (MOME) of Parameters
Since the QLD (1.3) 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 QLD (1.3), we have

Maximum Likelihood Estimate (MLE) of Parameters
Let ( ) The natural log likelihood function is thus obtained as where x is the sample mean.
The two log likelihood equations are obtained as

Goodness of Fit to Real Lifetime Data-Sets
The quasi Lindley distribution (QLD) has been fitted to a number of lifetime data-sets. In this section, we present the fitting of QLD to five real lifetime data-sets and compare its goodness of fit with exponential and Lindley distributions. The following five lifetime data-sets have been considered for comparing the goodness of fit of QLD with Lindley and exponential distributions.

Concluding Remarks
In the this paper some important mathematical and statistical properties including moment generating function, mean deviations about mean and median, order statistics, Bonferroni and Lorenz curves, Renyi entropy measure and stress strength reliability of quasi Lindley distribution (QLD) of Shanker and Mishra (2013 a) have been derived and discussed. The distribution has been fitted to some real lifetime data-sets to test its goodness of fit over exponential and Lindley distributions. It is clear from the fitting of QLD that it gives better fitting than exponential and Lindley distributions and hence QLD can be recommended over exponential and Lindley distributions for modeling real lifetime data-sets from biomedical science and engineering.