A Compound Poisson Perspective of Ewens–Pitman Sampling Model

Emanuele Dolera and Stefano Favaro
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Emanuele Dolera: Department of Mathematics, University of Pavia, Via Adolfo Ferrata 5, 27100 Pavia, Italy
Stefano Favaro: Collegio Carlo Alberto, Piazza V. Arbarello 8, 10122 Torino, Italy

Mathematics, 2021, vol. 9, issue 21, 1-12

Abstract: The Ewens–Pitman sampling model (EP-SM) is a distribution for random partitions of the set { 1 , … , n } , with n ? N , which is indexed by real parameters ? and ? such that either ? ? [ 0 , 1 ) and ? > ? ? , or ? < 0 and ? = ? m ? for some m ? N . For ? = 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 ? ? ( 0 , 1 ) , or ? < 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 ? diversity. We discuss the proposed results and conjecture that analogous compound Poisson representations may hold for the class of ? -stable Poisson–Kingman sampling models—of which the EP-SM is a noteworthy special case.

Keywords: Berry–Esseen type theorem; Ewens–Pitman sampling model; exchangeable random partitions; log-series compound poisson sampling model; Mittag–Leffler distribution function; negative binomial compound poisson sampling model; Pitman’s ?-diversity; wright distribution function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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