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April 2020 A Berry–Esseen theorem for Pitman’s α-diversity
Emanuele Dolera, Stefano Favaro
Ann. Appl. Probab. 30(2): 847-869 (April 2020). DOI: 10.1214/19-AAP1518

Abstract

This paper contributes to the study of the random number Kn of blocks in the random partition of {1,,n} induced by random sampling from the celebrated two parameter Poisson–Dirichlet process. For any α(0,1) and θ>α Pitman (Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that nαKna.s.Sα,θ as n+, where the limiting random variable, referred to as Pitman’s α-diversity, is distributed according to a polynomially scaled Mittag–Leffler distribution function. Our main result is a Berry–Esseen theorem for Pitman’s α-diversity Sα,θ, namely we show that supx0|P[Knnαx]P[Sα,θx]|C(α,θ)nα holds for every nN with an explicit constant term C(α,θ), for α(0,1) and θ>0. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of Kn in terms of a compound distribution; (ii) a quantitative version of the Laplace’s approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry–Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.

Citation

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Emanuele Dolera. Stefano Favaro. "A Berry–Esseen theorem for Pitman’s -diversity." Ann. Appl. Probab. 30 (2) 847 - 869, April 2020. https://doi.org/10.1214/19-AAP1518

Information

Received: 1 December 2018; Revised: 1 July 2019; Published: April 2020
First available in Project Euclid: 8 June 2020

zbMATH: 07236136
MathSciNet: MR4108124
Digital Object Identifier: 10.1214/19-AAP1518

Subjects:
Primary: 60F15 , 60G57

Keywords: Berry–Esseen theorem , Ewens–Pitman sampling model , Laplace approximation of integrals , Pitman’s -diversity , Poisson approximation , Poisson compound random partitions

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.30 • No. 2 • April 2020
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