The Poisson X-exponential Distribution: Theory, Estimation, and Applications in CountData Modeling
The Poisson Xexponential Distribution
Abstract
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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