A one-parameter poisson-emrem model for count data: Theory and applications to cytogenetic and COVID-19 studies

This paper proposes the Poisson-Emrem (PEm) distribution, a new one-parameter discrete probability model obtained by compounding the Poisson and Emrem distributions. The model preserves the mathematical tractability of the Poisson law while offering greater flexibility for analyzing count data in health and biological sciences. A theoretical study establishes its principal properties, showing that increases in the parameter lead to systematic reductions in the mean, variance, and interquartile range, with a corresponding increase in skewness. Additional features such as moments, dispersion index, and the hazard rate function are derived, the latter characterized by a bathtub-shaped form that is particularly suitable for lifetime studies. Parameters are estimated using maximum likelihood, and Monte Carlo simulations confirm the accuracy and efficiency of the estimator. The applicability of the PEm distribution is demonstrated through two empirical datasets involving cytogenetic abnormalities in rabbit lymphoid cells under streptogramin treatment and daily COVID-19 mortality counts in South Korea. Comparative goodness-of-fit analyses show that the proposed model consistently provides superior performance over competing discrete distributions. Owing to its ability to effectively capture skewness, over-dispersion, and zero inflation, the PEm distribution offers a flexible and reliable framework for modeling complex count data arising in health and biological sciences. These findings highlight its potential usefulness in public health monitoring and related statistical applications.