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

TL;DR

This paper analyzes the distribution of ordered and size-biased frequencies in GEM and Gibbs models for species sampling, extending known identities and providing new constructions for sampling and frequency ordering.

Contribution

It generalizes a known distribution identity for GEM models to include the two-parameter case and extends the description to Gibbs frequencies with new sampling constructions.

Findings

Describes the distribution of ordered frequencies in GEM and Gibbs models.

Extends known identities to the two-parameter GEM case.

Provides new sampling and frequency ordering constructions.

Abstract

We describe the distribution of frequencies ordered by sample values in a random sample of size $n$ from the two parameter 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 Tavar\'e (1986) for $\alpha = 0$ to the case $0 \le \alpha < 1$. This description extends to sampling from Gibbs$(\alpha)$ frequencies obtained by suitable conditioning of the GEM$(\alpha,\theta)$ model, and yields a value-ordered version of the Chinese Restaurant construction of GEM$(\alpha,\theta)$ and 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.