A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data

In applications including insurance claim counts, epidemiology, reliability analysis, finance, and network traffic modeling, count data with over dispersion and heavy-tailed behavior are common. Because they impose exponentially decaying tails that underestimate extreme count probabilities, classical Poisson and Poisson–Gamma (Negative Binomial) models are frequently insufficient in such situations. In this study, we propose a generalized Poisson–Gamma hybrid distribution obtained by compounding a Poisson distribution with a power–Gamma mixing law. The proposed model extends the classical Poisson–Gamma framework by introducing an additional shape parameter that governs tail thickness and induces greater dispersion. An explicit infinite-series representation of the probability mass function is derived, and fundamental distributional properties are investigated. It was shown that the Negative Binomial distribution arises as a special case, ensuring model coherence. Furthermore, the proposed model exhibits heavier-than-exponential tails, and under suitable parameter regimes, its tail probabilities display polynomial decay, placing the distribution within the class of heavy-tailed count models. Estimation and inferential aspects were discussed and applications to simulated count data and Monte Carlo simulated count data were also discussed. Heavy-tailed count data demonstrate superior performance compared to classical alternatives.