Some Poisson mixtures distributions with a hyperscale parameter

Abstract

We mainly investigate certain mixtures of Poisson distributions with a scale parameter in the mixing distribution. They help us to derive the bivariate Poisson mixtures arising from the prior and posterior predictive distributions in the semi-conjugate family defined by Laurent and Legrand (ESAIM Probab. Stat. (2011) DOI:10.1051/ps/2010018) for the “two Poisson samples” model, which contains in particular the reference prior when the parameter of interest is the ratio of the two Poisson rates.