On a General Class of Discrete Bivariate Distributions

In this paper we develop a general class of bivariate discrete distributions. The basic idea is quite simple. The marginals are obtained by taking the random geometric sum of the baseline random variables. The proposed class of distributions is a flexible class of bivariate discrete distributions in the sense the marginals can take variety of shapes. The probability mass functions of the marginals can be heavy tailed, unimodal as well as multimodal. It can be both over dispersed as well as under dispersed. We discuss different properties of the proposed class of bivariate distributions. The proposed distribution has some interesting physical interpretations also. Further, we consider two specific base line distributions: Poisson and negative binomial distributions for illustrative purposes. Both of them are infinitely divisible. The maximum likelihood estimators of the unknown parameters cannot be obtained in closed form. They can be obtained by solving three and five dimensional non-linear optimizations problems, respectively. To avoid that we propose to use expectation maximization algorithm, and it is observed that the proposed algorithm can be implemented quite easily in practice. We have performed some simulation experiments to see how the proposed EM algorithm performs, and it works quite well in both the cases. The analysis of one real data set has been performed to show the effectiveness of the proposed class of models. Finally, we discuss some open problems and conclude the paper.