Bivariate binomial conditionals distributions with positive and negative correlations: A statistical study

TL;DR

This paper introduces a new bivariate distribution with binomial conditionals that can model both positive and negative correlations, explores its properties, and demonstrates its application to real data.

Contribution

It presents a novel bivariate distribution with binomial conditionals, analyzes its properties, and shows its applicability to negative correlation data sets.

Findings

Distribution exhibits negative correlation and over-dispersion.

Distribution belongs to the exponential family.

Successfully fitted to real bivariate count data.

Abstract

In this article, we discuss a bivariate distribution whose conditionals are univariate binomial distributions and the marginals are not binomial that exhibits negative correlation. Some useful structural properties of this distribution namely marginals, moments, generating functions, stochastic ordering are investigated. Simple proofs of negative correlation, marginal over-dispersion, distribution of sum and conditional given the sum are also derived. The distribution is shown to be a member of the multi-parameter exponential family and some natural but useful consequences are also outlined. The proposed distribution tends to a recently investigated conditional Poisson distribution studied by Ghosh et al. (2020). Finally, the distribution is fitted to two bivariate count data sets with an inherent negative correlation to illustrate its suitability.