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A Compound Poisson Perspective of Ewens–Pitman Sampling Model

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  • Emanuele Dolera

    (Department of Mathematics, University of Pavia, Via Adolfo Ferrata 5, 27100 Pavia, Italy
    Collegio Carlo Alberto, Piazza V. Arbarello 8, 10122 Torino, Italy
    IMATI-CNR “Enrico Magenes”, 27100 Pavia, Italy)

  • Stefano Favaro

    (Collegio Carlo Alberto, Piazza V. Arbarello 8, 10122 Torino, Italy
    IMATI-CNR “Enrico Magenes”, 27100 Pavia, Italy
    Department of Economic and Social Sciences, Mathematics and Statistics, University of Torino, Corso Unione Sovietica 218/bis, 10134 Torino, Italy)

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 α and θ such that either α ∈ [ 0 , 1 ) and θ > − α , or α < 0 and θ = − m α for some m ∈ N . 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.

Suggested Citation

  • Emanuele Dolera & Stefano Favaro, 2021. "A Compound Poisson Perspective of Ewens–Pitman Sampling Model," Mathematics, MDPI, vol. 9(21), pages 1-12, November.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2820-:d:673380
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    References listed on IDEAS

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    1. Charalambos A. Charalambides, 2007. "Distributions of Random Partitions and Their Applications," Methodology and Computing in Applied Probability, Springer, vol. 9(2), pages 163-193, June.
    2. Stefano Favaro & Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2009. "Bayesian nonparametric inference for species variety with a two parameter Poisson-Dirichlet process prior," Carlo Alberto Notebooks 123, Collegio Carlo Alberto.
    3. Stefano Favaro & Antonio Lijoi & Ramsés H. Mena & Igor Prünster, 2009. "Bayesian non‐parametric inference for species variety with a two‐parameter Poisson–Dirichlet process prior," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 993-1008, November.
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