A Poisson-Generalized Mixture Exponential Distribution

Abstract

In this study, a new Poisson-generalized exponential distribution is proposed. Its statistical properties, including the cumulative distribution function, probability generating function, moment generating function, survival function, and index of dispersion, are derived. The proposed distribution has three parameters and exhibits a right-skewed, unimodal shape. Parameter estimation is performed using the maximum likelihood method. The distribution is then applied to two real overdispersed count datasets and compared with the Poisson and negative binomial distributions. The results indicate that the Poisson-generalized exponential distribution provides a better fit to both datasets based on the Akaike information criterion, Bayesian information criterion, negative log-likelihood, and the Kolmogorov–Smirnov goodness-of-fit test. These findings suggest that it is a viable alternative for modeling overdispersed count data. Future research may explore its extension to regression frameworks and other applied contexts.