TY - JOUR

T1 - Negative binomial construction of random discrete distributions on the infinite simplex

AU - Ipsen, Yuguang F.

AU - Maller, Ross A.

N1 - Publisher Copyright:
© 2017 Ukrainian National Academy of Sciences. All rights reserved.

PY - 2017

Y1 - 2017

N2 - The Poisson-Kingman distributions, PK(ρ), on the infinite simplex, can be constructed from a Poisson point process having intensity density ρ or by taking the ranked jumps up till a specified time of a subordinator with Lévy density ρ, as proportions of the subordinator. As a natural extension, we replace the Poisson point process with a negative binomial point process having parameter r > 0 and Lévy density ρ, thereby defining a new class PK(r)(ρ) of distributions on the infinite simplex. The new class contains the two-parameter generalisation PD(α, θ) of [13] when θ > 0. It also contains a class of distributions derived from the trimmed stable subordinator. We derive properties of the new distributions, with particular reference to the two most well-known PK distributions: the Poisson-Dirichlet distribution PK(ρθ) generated by a Gamma process with Lévy density ρθ(x) = θe−x/x, x > 0, θ > 0, and the random discrete distribution, PD(α, 0), derived from an α-stable subordinator.

AB - The Poisson-Kingman distributions, PK(ρ), on the infinite simplex, can be constructed from a Poisson point process having intensity density ρ or by taking the ranked jumps up till a specified time of a subordinator with Lévy density ρ, as proportions of the subordinator. As a natural extension, we replace the Poisson point process with a negative binomial point process having parameter r > 0 and Lévy density ρ, thereby defining a new class PK(r)(ρ) of distributions on the infinite simplex. The new class contains the two-parameter generalisation PD(α, θ) of [13] when θ > 0. It also contains a class of distributions derived from the trimmed stable subordinator. We derive properties of the new distributions, with particular reference to the two most well-known PK distributions: the Poisson-Dirichlet distribution PK(ρθ) generated by a Gamma process with Lévy density ρθ(x) = θe−x/x, x > 0, θ > 0, and the random discrete distribution, PD(α, 0), derived from an α-stable subordinator.

KW - Mixing distribution

KW - Poisson-Dirichlet distribution

KW - Poisson-Kingman distribution

KW - Size-biased constructions

KW - Stick-breaking

KW - Trimmed α-stable subordinator

UR - http://www.scopus.com/inward/record.url?scp=85053864885&partnerID=8YFLogxK

M3 - Article

SN - 0321-3900

VL - 22

SP - 34

EP - 46

JO - Theory of Stochastic Processes

JF - Theory of Stochastic Processes

IS - 2

ER -