Statistical inference for distributions with one Poisson conditional

It will be recalled that the classical bivariate normal distributions have normal marginals and normal conditionals. It is natural to ask whether a similar phenomenon can be encountered involving Poisson marginals and conditionals. However, it is known, from research on conditionally specified models, that Poisson marginals will be encountered, together with both conditionals being of the Poisson form, only in the case in which the variables are independent. In order to have a flexible dependent bivariate model with some Poisson components, in the present article, we will be focusing on bivariate distributions with one marginal and the other family of conditionals being of the Poisson form. Such distributions are called Pseudo-Poisson distributions. We discuss distributional features of such models, explore inferential aspects and include an example of applications of the Pseudo-Poisson model to sets of over-dispersed data.