  Considering discrete models, the univariate framework has been studied in
depth compared to the multivariate one. This paper first proposes two criteria
to define a sensu stricto multivariate discrete distribution. It then
introduces the class of splitting distributions that encompasses all usual
multivariate discrete distributions (multinomial, negative multinomial,
multivariate hypergeometric, multivariate neg- ative hypergeometric, etc . . .
) and contains several new. Many advantages derive from the compound aspect of
split- ting distributions. It simplifies the study of their characteris- tics,
inferences, interpretations and extensions to regression models. Moreover,
splitting models can be estimated only by combining existing methods, as
illustrated on three datasets with reproducible studies.
