  Count data are omnipresent in many applied fields, often with overdispersion
due to an excess of zeroes or extreme values. With mixtures of Poisson
distributions representing an elegant and appealing modelling strategy, we
focus here on the challenging problem of identifying a suitable mixing
distribution and study how extreme value theory can be used. We propose an
original strategy to select the most appropriate candidate among three
categories: Fr{\'e}chet, Gumbel and pseudo-Gumbel. Such an approach is
presented with the aid of a decision tree and evaluated with numerical
simulations.
