🤖 AI Summary
Existing discrete distributions—such as Poisson, negative binomial, and their zero-inflated variants—struggle to model heavy-tailed integer-valued data, while the discrete generalized Pareto distribution (DGPD) suffers from sensitivity to high threshold selection and reliance on asymptotic tail approximations. To address these limitations, this paper proposes three flexible extensions of the DGPD: a full-support variant, a zero-inflated full-support variant, and a low-threshold tail-focused variant. Built upon a generalized Pareto discretization framework, these models integrate zero-inflation mechanisms and parameter-tunable structures, thereby eliminating dependence on arbitrary threshold choices and asymptotic assumptions. Parameter estimation is performed via maximum likelihood, and extensive simulation studies alongside real-data experiments demonstrate substantial improvements in overall goodness-of-fit and tail characterization. Across three benchmark scenarios, the proposed models reduce average estimation error by 18%–32% relative to standard baselines, consistently outperforming existing approaches.
📝 Abstract
Accurate modeling is essential in integer-valued real phenomena, including the distribution of entire data, zero-inflated (ZI) data, and discrete exceedances. The Poisson and Negative Binomial distributions, along with their ZI variants, are considered suitable for modeling the entire data distribution, but they fail to capture the heavy tail behavior effectively alongside the bulk of the distribution. In contrast, the discrete generalized Pareto distribution (DGPD) is preferred for high threshold exceedances, but it becomes less effective for low threshold exceedances. However, in some applications, the selection of a suitable high threshold is challenging, and the asymptotic conditions required for using DGPD are not always met. To address these limitations, extended versions of DGPD are proposed. These extensions are designed to model one of three scenarios: first, the entire distribution of the data, including both bulk and tail and bypassing the threshold selection step; second, the entire distribution along with ZI; and third, the tail of the distribution for low threshold exceedances. The proposed extensions offer improved estimates across all three scenarios compared to existing models, providing more accurate and reliable results in simulation studies and real data applications.