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

Part of: Distribution theory Stochastic processes Applications

Published online by Cambridge University Press:  01 July 2021

Yuguang Ipsen ,
Ross A. Maller Open the ORCID record for Ross A. Maller [Opens in a new window]  and
Soudabeh Shemehsavar
Yuguang Ipsen*
Affiliation:
The Australian National University
Ross A. Maller*
Affiliation:
The Australian National University
Soudabeh Shemehsavar*
Affiliation:
University of Tehran
*
*Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT, 0200, Australia.
*Postal address: Research School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT, 0200, Australia.
**Postal address: School of Mathematics, Statistics & Computer Sciences, University of Tehran. Email address: Shemehsavar@ut.ac.ir
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Abstract

We derive the large-sample distribution of the number of species in a version of Kingman’s Poisson–Dirichlet model constructed from an $\alpha$ -stable subordinator but with an underlying negative binomial process instead of a Poisson process. Thus it depends on parameters $\alpha\in (0,1)$ from the subordinator and $r>0$ from the negative binomial process. The large-sample distribution of the number of species is derived as sample size $n\to\infty$ . An important component in the derivation is the introduction of a two-parameter version of the Dickman distribution, generalising the existing one-parameter version. Our analysis adds to the range of Poisson–Dirichlet-related distributions available for modeling purposes.

Keywords

Generalised Poisson–Dirichlet lawsnegative binomial point processnumber of speciesEwens sampling formulasize-biased samplinggeneralised Dickman distribution

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G52: Stable processes 60G55: Point processes
Secondary: 60G57: Random measures 62E20: Asymptotic distribution theory 62P10: Applications to biology and medical sciences
Type
Original Article
Information
Advances in Applied Probability , Volume 53 , Issue 2 , June 2021 , pp. 370 - 399
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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