Generalised Poisson-Dirichlet Distributions and the Negative Binomial Point Process

2016/11/30 by Ipsen, Yuguang F., Maller, Ross A.
#60G51 #60G52 #60G55 #60G57 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1611.09980

Abstract

When S=(S_t)_{t≥ 0} is an α-stable subordinator, the sequence of ordered jumps of S, up till time 1, omitting the r largest of them, and taken as proportions of their sum ^(r)S_t, defines a 2-parameter distribution on the infinite dimensional simplex, ∇_∞, which we call the PD_α^(r) distribution. When r=0 it reduces to the PD_α distribution introduced by Kingman in 1975. We observe a serendipitous connection between PD_α^(r) and the negative binomial point process of Gregoire (1984), which we exploit to analyse in detail a size-biased version of PD_α^(r). As a consequence we derive a stick-breaking representation for the process and a useful form for its distribution. This program produces a large new class of distributions available for a variety of modelling purposes.

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