A new bivariate Poisson distribution via conditional specification: properties and applications

In this article, we discuss a bivariate Poisson distribution whose conditionals are univariate Poisson distributions and the marginals are not Poisson which 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. Parameter estimation with maximum likelihood is implemented. Copula-based simulation experiments are carried out using Bivariate Normal and the Farlie–Gumbel–Morgenstern copulas to assess how the model behaves in dealing with the situation. Finally, the distribution is fitted to seven bivariate count data sets with an inherent negative correlation to illustrate suitability.