Modelling underdispersed count differences with a mean-parameterised CMP–Skellam model

This paper introduces a new generalisation of the Skellam distribution based on independent mean-parameterised Conway-Maxwell-Poisson (MPCMP) marginals with flexible, unequal dispersion. The resulting MPCMP-Skellam model can capture both under- and overdispersion – including bidispersion – in the difference of two count processes, which arise frequently in sports applications. We demonstrate the utility of this model through simulation studies and three real sports datasets: goal differences from Italy’s Serie A and Tunisia’s Ligue 1, and football yellow card differences. The proposed model offers an improvement over the Skellam model in each case, is implemented in reproducible code and extends the toolkit available to sports analysts for modelling count differences subject to less variability than permiited via adoption of the standard Skellam model.