A Bivariate Poisson-Gamma Distribution: Statistical Properties and Practical Applications
Indranil Ghosh, Mina Norouzirad, Filipe J. Marques
TL;DR
The paper defines a new bivariate mixed discrete-continuous distribution, the BPGC, via conditional specifications X|Y = y ~ Poisson(exp(m10 - m11 y + m12 log y)) and Y|X = x ~ Gamma(m02 + m12 x, m01 + m11 x y), with the joint density belonging to a five-parameter exponential family. It establishes key structural properties, including TP2 dependence, exponential-family representation with complete sufficient statistics, and stochastic-order results, along with non-linear regression behavior between X and Y. Parameter estimation is conducted through constrained maximum likelihood using adaptive barrier methods, supported by a Gibbs-based simulation study and a Fasano-Franceschini goodness-of-fit test to validate performance. The authors demonstrate practical applicability with a hospital admissions-costs dataset, achieving a good fit and illustrating the model’s capacity to capture mixed discrete-continuous phenomena. Overall, the work provides a theoretically rich and practically useful framework for modeling joint distributions with one discrete and one continuous component, and suggests avenues for multivariate extensions.
Abstract
Although the specification of bivariate probability models using a collection of assumed conditional distributions is not a novel concept, it has received considerable attention in the last decade. In this study, a bivariate distribution-the bivariate Poisson-Gamma conditional distribution-is introduced, combining both univariate continuous and discrete distributions. This work explores aspects of this model's structure and statistical inference that have not been studied before. This paper contributes to the field of statistical modeling and distribution theory through the use of maximum likelihood estimation, along with simulations and analyses of real data.