The Poisson–QGamma Distribution: Properties, Estimation Methods, Regression Modeling, and Applications in Engineering Count Data

Modeling over-dispersed count data is a common challenge in applied statistics, especially in engineering applications where repeated events, system faults, and clustered observations often produce variability beyond that allowed by the classical Poisson model. In this paper, we introduce and study the Poisson–QGamma distribution, a new compound discrete model obtained by mixing the Poisson distribution with the QGamma distribution. The proposed distribution is analytically tractable and flexible enough to capture over-dispersion, skewness, and excess kurtosis, which are frequently observed in real count data. Several statistical properties of the distribution are derived, including the probability mass function, cumulative distribution function, survival and hazard rate functions, moments, dispersion index, skewness, kurtosis, entropy, and generating functions. Parameter estimation is considered using maximum likelihood, method of moments, least squares, and weighted least squares methods. The finite-sample behavior of these estimators is examined through Monte Carlo simulation. A regression model based on the Poisson–QGamma distribution is also developed for count responses with covariates. The proposed model is compared with classical and competing count models using simulation and real-data applications. Three engineering-related datasets, involving power grid failure counts, environmental sensor event counts, and packet loss counts in communication networks, are analyzed to illustrate the practical value of the model. The results show that the Poisson–QGamma model provides a better fit than several standard alternatives, including the Poisson, negative binomial, Poisson–Lindley, generalized Poisson, and COM–Poisson models, particularly in the presence of over-dispersion and heavy-tailed behavior. Overall, the proposed distribution offers a parsimonious and effective tool for modeling over-dispersed count data, while also contributing to the broader class of compound discrete distributions.