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

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.

Generalized Poisson--Dirichlet Distributions Based on the Dickman Subordinator

“…(ii) Theorem 3.1 of Maller & Shemehsavar (2023) that a sampling model based on the Dickman subordinator has the same limiting behaviour as the Ewens sampling formula. This is no surprise as the Dickman function is well known to be closely associated with PD(θ) and some of its generalisations; see for example Arratia, Barbour & Tavaré (2003), pp.14, 76, Arratia & Baxendale (2015, Handa (2009), , 2021, Maller & Shemehsavar (2023).…”

“…From Eq. (5.11) of Ipsen, Maller & Shemehsavar (2021) (hereafter, "IMS") we get the following formula for the distribution of the (n+1)-vector (M n (α, r), K n (α, r)): (2.5)…”