A New Discrete Power Function Distribution with Its Mathematical Properties

This paper introduces a novel discrete probability distribution, termed the Discrete Power Function Distribution (DPFD), developed by discretizing the continuous power function distribution. Unlike traditional discrete models such as the Poisson, Geometric, Discrete Weibull, or Beta-Poisson distributions see [1,2,3,4,5]. The DPFD incorporates two shape-controlling parameters, enabling it to model a broader range of dispersion and skewness behaviors. The DPFD can exhibit flexible hazard rate functions, including increasing, decreasing, and bathtub shapes features that are uncommon in classical models. We rigorously derive its fundamental properties, including moments, quantiles, and order statistics, and propose maximum likelihood estimation methods for its parameters. Comparative analysis using real-world count data reveals that the DPFD consistently outperforms the Discrete Weibull and Beta-Poisson distributions, particularly in datasets characterized by over-dispersion and heavy tails, thus highlighting its superior adaptability and modeling power.