On a General Class of Discrete Bivariate Distributions

TL;DR

This paper introduces a highly flexible class of bivariate discrete distributions based on geometric sums, capable of modeling diverse data shapes, with applications demonstrated through real data analysis.

Contribution

It develops a general framework for bivariate discrete distributions with flexible marginals, including specific cases using Poisson and negative binomial distributions.

Findings

Distributions can be multimodal and heavy-tailed.

The class allows a wide range of correlation structures.

Method of moments estimators are effective for parameter estimation.

Abstract

In this paper we develop a very general class of bivariate discrete distributions. The basic idea is very simple. The marginals are obtained by taking the random geometric sum of a baseline distribution function. The proposed class of distributions is a very flexible class of distributions in the sense the marginals can take variety of shapes. It can be multimodal as well as heavy tailed also. It can be both over dispersed as well as under dispersed. Moreover, the correlation can be of a wide range. We discuss different properties of the proposes class of bivariate distributions. The proposed distribution has some interesting physical interpretations also. Further, we consider two specific base line distributions namely; 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 the method of moment estimators and they can be obtained quite conveniently. The analyses of two real data sets have been performed to show the effectiveness of the proposed class of models. Finally, we discuss some open problems and conclude the paper.