Ordered and size-biased frequencies in GEM and Gibbs’ models for species sampling

Abstract

We describe the distribution of frequencies ordered by sample values in a random sample of size $n$ from the two parameter $\mathsf{GEM}(\alpha,\theta)$ random discrete distribution on the positive integers. These frequencies are a (size-$\alpha$)-biased random permutation of the sample frequencies in either ranked order, or in the order of appearance of values in the sampling process. This generalizes a well-known identity in distribution due to Donnelly and Tavare [Adv. in Appl. Probab. 18 (1986) 1–19] for $\alpha=0$ to the case $0\le\alpha<1$. This description extends to sampling from $\operatorname{Gibbs}(\alpha)$ frequencies obtained by suitable conditioning of the $\mathsf{GEM}(\alpha,\theta)$ model, and yields a value-ordered version of the Chinese restaurant construction of $\mathsf{GEM}(\alpha,\theta)$ and $\operatorname{Gibbs}(\alpha)$ frequencies in the more usual size-biased order of their appearance. The proofs are based on a general construction of a finite sample $(X_{1},\dots,X_{n})$ from any random frequencies in size-biased order from the associated exchangeable random partition $\Pi_{\infty}$ of $\mathbb{N}$ which they generate.