Robustness of statistical methods for modeling paired count data using bivariate discrete distributions with general dependence structures

Bivariate Poisson models are appropriate for modeling paired count data. However the bivariate Poisson model does not allow for negative dependence structure, therefore it is necessary to consider alternatives, which can produce both positive and negative dependence. 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 structure, any copula misspecification can render the inference useless. In this work, we consider bivariate Poisson models generated with a copula and investigate its robustness under outliers contamination and model misspecification. Particular focus is given on the robustness of copula related parameters.