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 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 Tavaré (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.

23 pages, some corrections to the first version

References in corpus (3)

• Gibbs Partitions (EPPF's) Derived From a Stable Subordinator are Fox H and Meijer G Transforms
• Wright-Fisher construction of the two-parameter Poisson-Dirichlet diffusion
• Looking-backward probabilities for Gibbs-type exchangeable random partitions

Cited by in corpus (1)

• An ergodic theorem for partially exchangeable random partitions