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We describe the distribution of frequencies ordered by sample values in a random sample of size n from the two parameter GEM(a, ?) random discrete distribution on the positive integers. These frequencies are a (size-a)-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 a = 0 to the case 0 = a < 1. This description extends to sampling from Gibbs(a) frequencies obtained by suitable conditioning of the GEM(a, ?) model, and yields a value-ordered version of the Chinese restaurant construction of GEM(a, ?) and Gibbs(a) frequencies in the more usual size-biased order of their appearance. The proofs are based on a general construction of a finite sample (X1, . . ., Xn) from any random frequencies in size-biased order from the associated exchangeable random partition 8 of N which they generate.
Original language | English |
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Pages (from-to) | 1793-1820 |
Number of pages | 28 |
Journal | Annals of Applied Probability |
Volume | 28 |
Issue number | 3 |
DOIs | |
State | Published - 1 Jun 2018 |
ID: 32734430