International Journal of Computer Applications (0975 – 8887) Volume 95– No. 13, June 2014 9 Quasi Lindley Geometric Distribution L.S. Diab College of Science for (girls) Dept. of Mathematics, Al-Azhar University Nasr City, 11884, Egypt Hiba Z. Muhammed Institute of Statistical Studies and Research Department of Mathematical Statistics Cairo University, Cairo, Egypt ABSTRACT In this paper, we introduce a new class of lifetime distributions which is called the Quasi Lindley Geometric (QLG) distribution. This distribution obtained by compounding the Quasi Lindley and geometric distributions. Some structural properties of the proposed new distribution are discussed, including probability density function and explicit algebraic formulas for its survival and hazard functions, moment , moment generating function and mean deviations. We propose the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. A real data set is used to illustrate the importance and flexibility of the new distribution. Keywords Quasi Lindley distribution, Geometric distribution, Moments, Maximum likelihood 1. INTRODUCTION The Lindley distribution was introduced by Lindley (1958) as a new distribution useful to analyze lifetime data especially in applications modeling stress-strength reliability. In a recent paper Ghitany et al. (2008) studied the properties of the Lindley distribution under a carefully mathematical treatment. They also showed in a numerical example that the Lindley distribution gives better modeling for waiting times and survival times data than the exponential distribution. The use of the Lindley distribution could be a good alternative to analyse lifetime data within the competing risks approach as compared with the use of standard Exponential or even the Weibull distribution commonly used in this area. The Exponential distribution assumes constant hazard function, usually not an appropriated assumption for many competing risks data. Mazucheli and Achcarb (2011) studied the applications of Lindley distribution to competing risks lifetime data. Also, the Lindley distribution has some nice properties to be used in lifetime data analysis as closed forms for the survival and hazard functions and good flexibility of fit. Lindley (1958), introduced a one- parameter distribution, known as Lindley distribution, given by its probability density function 2 ( ,) ( ,) (1 ) ; 0, 0, 1 x f x gx xe x              the cumulative distribution function (cdf) of Lindley distribution is obtained as ( ,) 1 1 , 0, 0. 1 x x Fx e x                   Rama and Mishra (2013) introduced a new two-parameter Quasi Lindley distribution (QLD), of which the Lindley distribution (LD) is a particular case. They studied several properties of the QLD, and shown that the QLD is more flexible than Lindley and exponential distributions. Quasi Lindley distribution with parameters α and θ is defined by its probability density function (p.d.f) ( ,, ) ( ) ; 0, 0, 1. 1 x f x xe x                It can easily be seen that at  , the Equation (1.3) reduces to the Lindley distribution. And at  , it reduces to the gamma distribution with parameters  . The p.d.f. Equation (1.3) can be shown as a mixture of exponential  and gamma . distributions as follows 1 2 (,, ) () (1 ) () f x pg x pg x     where 2 1 2 1 , ( ) and ( ) . 1 x x p g x e g x e            The cumulative distribution function (cdf) of QLD is obtained as ( ,, ) 1 1 , 0, 0, 1. 1 x x Fx e x                     where  is scale parameter. Adamidis and Loukas (1998) introduced a two-parameter lifetime distribution with decreasing failure rate by compounding exponential and geometric distributions, which was named exponential geometric (EG) distribution. In the same way, the exponential Poisson (EP) and exponential logarithmic (EL) distributions were introduced and studied by Kus (2007) and Tahmasbi and Rezaei (2008), respectively. Recently, Chahkandi and Ganjali (2009) proposed a class of distributions, named exponential power series (EPS) distributions, by compounding exponential and power series distributions, where compounding procedure follows the same way that was previously carried out by Adamidis and Loukas (1998). In the same way, Barreto-Souza et al. (2010) and Lu and Shi (2011) introduced the Weibull-geometric (WG) and Weibull-Poisson (WP) distributions which naturally extend the EG and EP distributions, respectively. Barreto et al. (2009) presented a generalization of the exponential-Poisson distribution. Morais and Barreto-Souza (2011) defined the Weibull power series (WPS) class of distributions which contains the (EPS) distributions as sub-models. Adamidis et al. (2005) proposed the extended exponential-geometric (EEG) distribution which generalizes the EG distribution and discussed several of its statistical properties along with its reliability features. In this paper we introduce a Quasi Lindley-geometric (QLG) distribution which generalizes the Geometric and Quasi Lindley distributions and study some of its properties. The paper is organized as follows. In Section 2, we define the QLG distribution, density, hazard rate function and survival function. In Section 3, we give some statistical properties of the new distribution. The order statistics and its moment are given in Section 4. Residual life and reversed residual life functions of QLG distribution are discussed in Section 5. Mean deviations from the mean and median are derived in Section 6. The least squares and weighted least squares estimators are introduced in Section 7. In Section 8, we demonstrate the maximum likelihood estimates of the unknown parameters.