The Poisson X-exponential distribution: theory, estimation, and applications in count data modeling
Abstract
This paper revisits the Poisson Xexponential Distribution (PXED), a flexible discrete model for count data derived as a special case of a more general distributional framework. Although the PXED is not entirely new, its structure offers notable advantages in modeling moderate to high overdispersion and skewness—features often encountered in empirical count data. We derive the distribution’s key statistical properties, including its moments, skewness, kurtosis, and reliability functions. A suite of estimation techniques is considered, including Maximum Likelihood Estimation (MLE), Ordinary Least Squares (OLS), and Bayesian inference. The finite-sample performance of these estimators is assessed through extensive Monte Carlo simulations. To enhance practical relevance, the PXED is further embedded within a regression framework to accommodate covariate-dependent count outcomes. Model comparisons against standard and compound alternatives—such as the Poisson, Geometric, Negative Binomial, Poisson-Lindley, and Poisson-X-Lindley distributions —demonstrate that PXED yields competitive or superior fit in terms of log-likelihood, AIC, and BIC. Applications to real datasets on Epileptic Seizure Counts and insurance claims illustrate the model’s empirical effectiveness, particularly under MLE, OLS, and Bayesian estimation. These results highlight PXED as a valuable and robust option for flexible count data modeling in applied statistics.
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