A NEW GENERALIZATION OF LINDLEY DISTRIBUTION

In this study, we introduce a new generalization called the Lomax-Lindley distribution of Lindley distribution constructed by combining the cumulative distribution function (cdf) of Lomax and Lindley distributions. Some mathematical properties of the new distribution are discussed including moments, quantile and moment generating function. Estimation of the model parameters is carried out using maximum likelihood method. Finally, real data examples are presented to illustrate the usefulness and applicability of this new distribution.


Introduction
The Lindley distribution is another life time probability distribution which can be used in modeling data reliability, biology, finance and lifetime analysis. It was introduced by Lindley These distributions are modification, extension, or combinations of existing one. In this work we will concern with the last idea of combination of existing one. Gupta et al. (2016) suggested a new obtaining family of new distributions from two cdf F and G of known distributions with the following cdf: and the corresponding pdf as

2 Lomax-Lindley Distribution
In this section we studied the Lomax-Lindley (Lomax-L) distribution. Let F be the cdf The corresponding pdf of Lomax-Lindley distribution is given by The hazard function (hf) can be obtained using

Moments
The rth non-central moments of the Lomax-L distribution, denoted by , is given by the following theorem.
Theorm 1. If X is a continuous random variable has the Lomax-Lindley distribution, then the rth non-central moments is given by Proof: Let X be a random variable with density function (4). The rth non-central moment of Lomax-L distribution is given by Using binomial expansion, For any real non-integer b>0, and |z|<1, Gradshteyn and Ryzhik (2007) defined the power series: Using this fact, also by using binomial expansion, we get

Moment Generating Function
The moment generating function of the Lomax-L distribution, denoted by given by the following theorem.

Order Statistics
Let 1 , 2 , … , be a simple random sample from the Lomax-L distribution with cdf and pdf given by (3) and (4), respectively. Let (1) , (2) , … , ( ) denote the order statistics obtained from this sample. The pdf of ( ) , = 1,2, … , is given by Substituting from (3) and (4) into (5), we can get the pdf of ( ) . The moments of the order statistics of the Lomax-L distribution can be easily written in terms of moments of the Lomax-L distribution.

Maximum Likelihood Estimation
Let 1 , 2 , … , be a random sample of size n follows the Lomax-L distribution. To determine the maximum likelihood estimates (MLEs) of the three unknown parameters ( , , ) of the Lomax-L distribution we first obtain the likelihood function as follows: The log-likelihood function becomes: ln = ln + ln + ln 2 − ln − ln( + 1) + ∑ ln The first partial derivatives of the log-likelihood function with respect to, β α and θ are Setting these equations to zero and solving the resulting system of non-linear equations to obtain the MLEs of the three unknown parameters, α β and θ of the Lomax-L distribution, (̂,̂,̂). The negative second partial derivatives of the loglikelihood function are:  Table   (1). The estimated variance-covariance matrix for the Lomax-L distribution is given by:  Table (2). Results in Table 2 , indicates that the Lomax-L model gives the smallest -2lnL, AIC, BIC and CAIC and gives the second smallest KS value when compare to other distributions. We conclude that the lomax-L distribution is a better model than the other competitive models.

Conclusion
In this paper, we concerned with one of the new family of distributions suggested by