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.

Multivariate discrete distributions via sums and shares

“…. , M d ), previously arising in the bivariate case as a Bayesian predictive distribution [16]. Two contrasting applications of the latter model were investigated.…”

“…When d = 2, this is the Bailey distribution of Laurent [16] and the distribution in (4) thus represents its multivariate extension. The distribution of (R 1 , .…”

“…Here, 2 F 1 is the Gauss hypergeometric function. A closely related expression was given when d = 2 in [16]. For a pair of random variables, alongside the covariance and correlation, it is also of interest to consider local dependence which, in the case of discrete models, is measured by the set of log cross-product ratios of adjacent 2 × 2 cells in a (usually infinitely) large ordinal contingency table [10,11,25]…”

“…where P S is the survival function associated with p S . While p T can be specified arbitrarily on N 0 , (16) and (17) show that p S has to be a discrete ( . The most convenient form for multivariate moments of Schur constant distributions -which appears not to be in [8]…”

“…, U d following a Dirichlet distribution (so that M d−1 |T = t follows a Dirichlet-multinomial, or multivariate Pólya, distribution). The resulting joint distribution is a multivariate extension of what in the bivariate case Laurent [16] called the Bailey distribution. It is a focus of this article, and is considered in detail in Section 3.…”