A generalised Dickman distribution and the number of species in a negative binomial process model. (June 2021)

• Record Type: Journal Article
• Authors: Ipsen, Yuguang; Maller, Ross A.; Shemehsavar, Soudabeh
• Is Part Of: Advances in applied probability. Volume 53:Number 2(2021)
• Page Start: 370
• Page End: 399
• Publication Date: 2021-06
• Subjects: Generalised Poisson–Dirichlet laws -- negative binomial point process -- number of species -- Ewens sampling formula -- size-biased sampling -- generalised Dickman distribution; 60G51 -- 60G52 -- 60G55 -- 60G57 -- 62E20 -- 62P10; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2020.61
• Languages: English
• ISSNs: 0001-8678

Abstract

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.