A Structural Characterization of Hypergeometric-Type Discrete Distributions: Refinements, Tail Classifications, and Fractional Counting Applications

Abstract

We introduce a refined structural framework for discrete distributions whose probability mass functionsare generated by generalized hypergeometric series, expanding upon the classical Kemp families. The proposed class,termed the Generalized Hypergeometric Family Model (GHFM), is defined through hypergeometric normalizing constantsand accommodates both finite and infinite supports by allowing non-positive integers in the parameter vectors underexplicit truncation rules. We establish necessary and sufficient conditions under which a discrete distribution with arational probability ratio belongs to the GHFM class, thereby proving a rigorous structural characterization theorem. Theframework unifies a vast collection of classical laws, including the Poisson, negative binomial, binomial, and hypergeometricdistributions. As a primary modern application, we formally demonstrate how the discrete Fractional Poisson Distribution(FPD) derived via conformable fractional calculus emerges as a parameter-functionalized baseline special case withinthis unified architecture (p = 0, q = 0). We provide a comprehensive treatment of the FPD, including its probabilitymass function, cumulative distribution function, hazard rate function, factorial moments, probability generating function,quantiles, mean residual life, and R´enyi entropy. Additionally, we derive estimators using maximum likelihood, moments,least squares, Anderson–Darling, and Cram´er-von Mises methods. A comprehensive Monte Carlo simulation study isconducted to compare the finite-sample performance of the five estimation methods. The results demonstrate that themaximum likelihood estimator exhibits superior performance in terms of root mean squared error, while distance-basedmethods show systematic biases.We provide a rigorous mathematical diagnostic explaining this behavior: the discontinuousnature of the cumulative distribution function for discrete supports renders minimum-distance objective functions highly nondifferentiable,trapping gradient-based algorithms in biased local optima. The practical utility of the proposed framework isdemonstrated through a thorough real data analysis involving three well-known count datasets: school absenteeism counts,insurance claims, and scientific publication counts. The analysis reveals a critical insight: the estimated fractional parameterα consistently converges to the lower boundary, causing the FPD to collapse to the classical Poisson distribution. This finding,rather than indicating a model failure, validates the necessity of the broader GHFM framework. Specifically, it demonstratesthat the baseline special case (p = 0, q = 0) reaches its theoretical limit under extreme over-dispersion, mathematicallyjustifying the transition to higher-order GHFM configurations. The paper concludes with practical recommendationsfor applied researchers and outlines promising directions for future work, including multivariate extensions, Bayesianestimation, and the integration of Mittag-Leffler scaling for non-homogeneous fractional processes.