Samule Valiquette. 2020. Théorie des valeurs extrêmes dans le cadre des mélanges de Poisson.

• Abstract: In ecology, species distribution models are classically used to analyse the abundance and distribution of species. A solution consists in supposing that these observations are generated from a Poisson distribution. However several factors, biotics or abiotics, may induce overdispersion. This produces excess zeros or/and extremes values. Such situation violates the assumption that the expected value is equal to the variance. To overcome such a problem, mixed Poisson distributions provide a elegant solution. A huge set of mixture distribution is available. This work presents a strategy to shrink the choice of the possible distributions. It is based on an analysis of the tail behavior of the observations. Precisely, we propose to apply extreme value theory, usually used for continuous random variables, to Poisson mixtures. We introduce conditions that indicate when the domain of attraction of the mixed distribution is preserved or not by the final mixture. We also show that if a density with a 'gamma behavior' is used for the mixed distribution, then the mixture won't have any domain of attraction, but will be 'close' to the Gumbel domain. Such densities include the gamma and the inverse Gaussian. These results are used to established a decision tree based on the excess of the count data. We illustrate this strategy by applying it to an abundance data set of two common tree species from Central African rainforests.