Some Properties of GEM($\alpha$) Distributions

Abstract

A random discrete probability distribution known as the GEM($\alpha$) distribution, named after Griffiths, Engen and McCloskey, has found many applications for instance in population genetics, in mathematical ecology, in several probabilistic problems, in computer science etc. It reappeared under its current name “stick breaking distribution” in Sethuraman (Statist. Sinica, 4, 639–650, 1994) in the constructive definition of Dirichlet processes, which in turn has found numerous applications in Bayesian nonparametrics. The “invariant under size-biased property” (ISBP) of this distribution goes back to the Ph. D. dissertation of McCloskey (1965) and has been established by later authors (e.g. Donnelly J. Appl. Probab., 28, 321–335, 1991; Ongaro J. Statist. Plann. Inference, 128, 123–148, 2005; Pitman Adv. in Appl. Probab., 28, 525–539, 1996) by appealing to the Poisson Dirichlet distribution and random exchangeable permutations of the sets of integers. This paper gives a self contained proof of the ISBP property of stick breaking distribution using just its mixed moments. These mixed moments are also useful in nonparametric posterior estimation of the variance, skewness and kurtosis under Dirichlet process priors.