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Keywords = Pitman’s α-diversity

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12 pages, 308 KiB  
Article
A Compound Poisson Perspective of Ewens–Pitman Sampling Model
by Emanuele Dolera and Stefano Favaro
Mathematics 2021, 9(21), 2820; https://doi.org/10.3390/math9212820 - 6 Nov 2021
Cited by 3 | Viewed by 1873
Abstract
The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,,n}, with nN, which is indexed by real parameters α and θ such that either [...] Read more.
The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set {1,,n}, with nN, which is indexed by real parameters α and θ such that either α[0,1) and θ>α, or α<0 and θ=mα for some mN. For α=0, the EP-SM is reduced to the Ewens sampling model (E-SM), which admits a well-known compound Poisson perspective in terms of the log-series compound Poisson sampling model (LS-CPSM). In this paper, we consider a generalisation of the LS-CPSM, referred to as the negative Binomial compound Poisson sampling model (NB-CPSM), and we show that it leads to an extension of the compound Poisson perspective of the E-SM to the more general EP-SM for either α(0,1), or α<0. The interplay between the NB-CPSM and the EP-SM is then applied to the study of the large n asymptotic behaviour of the number of blocks in the corresponding random partitions—leading to a new proof of Pitman’s α diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of α-stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case. Full article
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