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Extreme sizes in Gibbs-type exchangeable random partitions

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  • Shuhei Mano

    (The Institute of Statistical Mathematics
    CREST, Japan Science and Technology Agency)

Abstract

Gibbs-type exchangeable random partition, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random combinatorial structures, and stochastic models of diversity in various phenomena. Some distributional results on ordered sizes in the Gibbs partition are established by introducing associated partial Bell polynomials and analysis of the generating functions. The combinatorial approach is applied to derive explicit results on asymptotic behavior of the extreme sizes in the Gibbs partition. Especially, Ewens–Pitman partition, which is the sample from the Poisson–Dirichlet process and has been discussed from rather model-specific viewpoints, and a random partition which was recently introduced by Gnedin, are discussed in the details. As by-products, some formulas for the associated partial Bell polynomials are presented.

Suggested Citation

  • Shuhei Mano, 2017. "Extreme sizes in Gibbs-type exchangeable random partitions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(1), pages 1-37, February.
  • Handle: RePEc:spr:aistmt:v:69:y:2017:i:1:d:10.1007_s10463-015-0530-0
    DOI: 10.1007/s10463-015-0530-0
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    References listed on IDEAS

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    1. Lijoi, Antonio & Mena, Ramses H. & Prunster, Igor, 2005. "Hierarchical Mixture Modeling With Normalized Inverse-Gaussian Priors," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 1278-1291, December.
    2. Antonio Lijoi & Igor Pruenster & Stephen G. Walker, 2008. "Bayesian nonparametric estimators derived from conditional Gibbs structures," ICER Working Papers - Applied Mathematics Series 06-2008, ICER - International Centre for Economic Research.
    3. Devroye, Luc, 1993. "A triptych of discrete distributions related to the stable law," Statistics & Probability Letters, Elsevier, vol. 18(5), pages 349-351, December.
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