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June 2018 Ordered and size-biased frequencies in GEM and Gibbs’ models for species sampling
Jim Pitman, Yuri Yakubovich
Ann. Appl. Probab. 28(3): 1793-1820 (June 2018). DOI: 10.1214/17-AAP1343


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 Tavaré [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.


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Jim Pitman. Yuri Yakubovich. "Ordered and size-biased frequencies in GEM and Gibbs’ models for species sampling." Ann. Appl. Probab. 28 (3) 1793 - 1820, June 2018.


Received: 1 April 2017; Revised: 1 August 2017; Published: June 2018
First available in Project Euclid: 1 June 2018

zbMATH: 06919738
MathSciNet: MR3809477
Digital Object Identifier: 10.1214/17-AAP1343

Primary: 60C05
Secondary: 60G09

Keywords: Chinese restaurant construction , GEM distribution , Gibbs’ partitions , random exchangeable partition , size-biased order , Species sampling

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.28 • No. 3 • June 2018
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