Abstract
This thesis focuses on certain theoretical aspects of counting data modeling. Two distinct frameworks are addressed: extreme values and multivariate modeling. Our first contribution explores, in terms of extreme behaviors, the existing connections between the Poisson mixture and its mixing distribution. This work allows us to characterize and discriminate several families of Poisson mixture according to their tail behavior. Although this work is theoretical, we discuss its practical utility, particularly regarding the choice of the mixing distribution. Our second contribution focuses on a new class of multivariate models called Tree Pólya Splitting. This class is based on hierarchical modeling and assumes that a random quantity is successively divided according to a Pólya distribution through a partition tree structure. In this work, we characterize univariate and multivariate marginal distributions, factorial moments, as well as the resulting dependency structures (covariance/correlation). Using a dataset corresponding to the abundance of Trichoptera, we highlight the interest of this class of models by comparing our results to those obtained, for example, with multivariate Poisson-lognormal models. We conclude this thesis by presenting various perspectives.