Modeling Overdispersed Football Goal Counts Using the Poisson-Tweedie Distribution

Abstract

Discrete Tempered Stable (DTS) distributions constitute a flexible class of probability models designed for over dispersed count data. By tempering the heavy tails of stable distributions, the DTS family ensures finite variance while retaining the ability to capture excess variability and skewness, making it well-suited for empirical count data. This paper reviews the fundamental properties of DTS distributions, including their probability generating function, dispersion indices, and connections to well-known subfamilies such as the Poisson–Tweedie and Poisson–inverse Gaussian distributions. An empirical study is conducted on 12,400 European football matches spanning the 2018–2024 seasons. The Poisson–Tweedie model a special case of the DTS class is estimated via maximum likelihood, yielding optimal parameter estimates: location parameter μ=2.585, dispersion parameter ϕ=0.966, and power parameter p=1.10. The fitted model substantially outperforms the standard Poisson distribution, particularly in capturing the upper tail corresponding to high-scoring matches. Model comparison using the Akaike Information Criterion (AIC) confirms the superiority of the Poisson–Tweedie model (AIC = 45,621) over the Poisson model (AIC = 46,197). The reduction of 576 units in AIC indicates a substantial improvement, equivalent to an exponential likelihood gain of exp(288) in favor of the Poisson–Tweedie model. From a methodological perspective, this work contributes to the development and application of advanced statistical tools for complex count data, aligning with the innovation-oriented aims of SDG 9 (Industry, Innovation and Infrastructure). These findings establish the Discrete Tempered Stable framework, and the Poisson–Tweedie subclass in particular, as a robust and reliable alternative to classical count models for over dispersed sports data.