Robustness methods for modelling count data with general dependence structures

Bivariate Poisson models are appropriate for modelling paired count data. However, the bivariate Poisson model does not allow for a negative dependence structure. Therefore, it is necessary to consider alternatives. A natural way is to consider copulas to generate various bivariate discrete distributions. While such models exist in the literature, the issue of choosing a suitable copula has been overlooked so far. Different copulas lead to different structures and any copula misspecification can render the inference useless. We consider bivariate Poisson models generated with a copula and investigate its robustness under outliers contamination and model misspecification. Particular focus is on the robustness of copula related parameters. English Premier League data are used to demonstrate the effectiveness of our approach.