Limiting Distributions of Generalised Poisson–Dirichlet Distributions Based on Negative Binomial Processes

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 (r) α distribution. When r = 0 it reduces to the PD α distribution introduced by Kingman in 1975. We observe a serendipitous connection between PD (r) α and the negative binomial point process of Gregoire (1984), which we exploit to analyse in detail a size-biased version of PD (r) α . 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.

Truncated two‐parameter Poisson–Dirichlet approximation for Pitman–Yor process hierarchical models

“…Let {J i } i≥1 be the ranked jumps of a stable process on the time interval [0, 1]. Denote by (N) 𝜏 1 the N-trimmed subordinator (Ipsen & Maller, 2017) derived from 𝜏 1 , that is, the process obtained by removing the N largest jumps from 𝜏 1 , such that (N) 𝜏 1 ∶= 𝜏 1 − ∑ N i=1 J i . The following lemma derives the conditional density of (N) 𝜏 1 .…”

A generalised Dickman distribution and the number of species in a negative binomial process model

“…Like , the class is based on an -stable subordinator, but the extra parameter r arises from a connection with the negative binomial point process introduced by Gregoire [9] (whereas is associated with a Poisson point process). Ipsen, Maller and Shemehsavar [14] developed connections between various Poisson–Dirichlet models by letting and in , while Ipsen, Shemehsavar and Maller [16] fitted to gene and species sampling data, demonstrating the utility of allowing the extra parameter r in a data analysis.…”

Ratios of ordered points of point processes with regularly varying intensity measures

“…Our aim in this paper is to give a systematic treatment of the limiting behaviour of ratios of ordered Poisson points. As is natural, we take a point process approach and make special connection with the negative binomial process whose relevance in the present context was brought out in Ipsen & Maller (2017b). This connection via ratios of points enabled the construction of a generalised kind of Poisson-Dirichlet distribution which can be added to the repertoire of available models for data analytic purposes.…”