ORDERED AND SIZE-BIASED FREQUENCIES IN GEM AND GIBBS' MODELS FOR SPECIES SAMPLING

• 成果类型: Article
• 署名作者: Pitman, Jim; Yakubovich, Yuri
• 署名单位: University of California System; University of California Berkeley; Saint Petersburg State University; Saint Petersburg State University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/17-AAP1343
• 发表日期: 2018
• 页码: 1793-1820
• 关键词: age; distributions; alleles; REPRESENTATION; reversibility; priors

摘要

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 Tavare [Adv. in Appl. Probab.18 (1986) 1-19] for alpha = 0 to the case 0 <= 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, . . . ,X-n) from any random frequencies in size-biased order from the associated exchangeable random partition Pi(infinity) of N which they generate.