The Mixed Weibull-Negative Binomial Distribution

In this paper, we introduce the Weibull Negative Binomial (WNB) with four parameters distribution which is obtained by compounding Weibull and Negative Binomial distributions, where compounding procedure follows same way that was previously carried out by Adamidis and Loukas (1998) and Kus (2007). This new distribution is a genaral case Exponential-geometric (EG) and Weibull-geometric distributions (WG), which was introduced recently by Adamidis and Loukas (1998) and Barreto-Souza et. all. (2010) respectively. The WNB distribution has increasing, decreasing, upside down bathtub hazard function. We obtain several properties of the WNB distributions such as moments, order statistics, estimation by maximum likelihood and inference for large sample. Furthermore, EM algorithm is also used to determine the maximum likelihood estimates of parameters and we discuss Renyi and Shannon entropy. Applications to real data sets are given to show the exibility and potentiality of the proposed distribution.