2017/01/23 by Jim Pitman, Pitman, Jim, Yu. V. Yakubovich +1
Computer Science · Mathematics · #60F05 #60G09 #60G70 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Statistical Distribution Estimation and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1701.06294
openalex publication_date 2017/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that in a sample of size n from a GEM(0,θ) random discrete distribution, the gaps Gi:n:= Xn-i+1:n - Xn-i:n between order statistics X1:n ≤ ⋯ ≤ Xn:n of the sample, with the convention Gn:n := X1:n - 1, are distributed like the first n terms of an infinite sequence of independent geometric(i/(i+θ)) variables Gi. This extends a known result for the minimum X1:n to other gaps in the range of the sample, and implies that the maximum Xn:n has the distribution of 1 + ∑i=1n Gi, hence the known result that Xn:n grows like θlog(n) as n→∞, with an asymptotically normal distribution. Other consequences include most known formulas for the exact distributions of GEM(0,θ) sampling statistics, including the Ewens and Donnelly--Tavaré sampling formulas. For the two-parameter GEM(α,θ) distribution we show that the maximal value grows like a random multiple of nα/(1-α) and find the limit distribution of the multiplier.