Multivariate mixed poisson distributions

Univariate Mixed Poisson distributions (MPDs) are commonly used to model data recorded from low flux objects or with short exposure times. They assume that the number of recorded events, conditioned on the received random intensity, is Poisson distributed. This communication focuses on the generalization of the MPDs to the multivariate case. This generalization is required to tackle new challenging problems such as exo-planet detection using direct imaging. The joint moments and the moment generating function of a multivariate mixed Poisson distribution (MMPD) are derived. These quantities allow to characterize the over-dispersion, dependency or unicity properties of the distribution. The important example of negative multinomial distributions is considered. These distributions are obtained when the mixing distribution is a multivariate Gamma distribution. Conditions ensuring that MMPDs belong to a natural exponential family (NEF) are finally investigated.