Marshall-Olkin Discrete Uniform Distribution

We introduce and characterize a new family of distributions, Marshall-Olkin discrete uniform distribution. The natures of hazard rate, entropy, and distribution of minimum of sequence of i.i.d. random variables are derived. First order autoregressive (AR (1)) model with this distribution for marginals is considered. The maximum likelihood estimates for the parameters are found out. Also, the goodness of the distribution is tested with real data. 1. Introduction Marshall and Olkin [1] introduced a new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Jose and Alice [2] discussed Marshall-Olkin family of distributions and their applications in time series modeling and reliability. Jose and Krishna [3] have developed Marshall-Olkin extended uniform distribution. These works and most of the references there in, deal with continuous distribution. Not much work is seen in the discrete case. The reason behind this may be that it is difficult to obtain compact mathematical expressions for moments, reliability, and estimation in the discrete set up. If is the survival function of a distribution, then, by Marshall-Olkin method, we get another survival function , by adding a new parameter to it. That is, Then the corresponding distribution function is . Now its probability mass function (p.m.f.) is where is the p.m.f. corresponding to . The hazard rate of is We consider a new family of distributions by adding an additional parameter by using the Marshall-Olkin scheme (Marshall and Olkin [1] in Section 2). The nature of hazard rate, entropy and expectation are derived. In the third section, distributions of minimum sequence of i.i.d. random variables are found out. An AR (1) model with new Marshall-Olkin discrete uniform distribution is discussed in the next section. In the fifth section, the maximum likelihood estimate (m.l.e.) for the parameters is found out and, in the last section, the goodness of the distribution is tested with a real life data. 2. Marshall-Olkin Discrete Uniform Distribution Let be a discrete random variable following uniform distribution with p.m.f. , . We have the distribution function and the survival function . By Marshall-Olkin method, we can form another survival function of Marshall-Olkin discrete uniform (MODU) distribution by substituting in , in (1), and we get We write for a random variable (r.v.) with given in (4). 2.1. Stability Property of the New Family If we apply the same method again into the new family, by adding a new parameter “ ” to the reliability