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Large sample asymptotics for the two-parameter Poisson–Dirichlet process

Lancelot F. James

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Abstract

This paper explores large sample properties of the two-parameter (α, θ) Poisson–Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension of the Dirichlet process, we explore the consistency and weak convergence of the the two-parameter Poisson–Dirichlet posterior process. We also establish the weak convergence of properly centered two-parameter Poisson–Dirichlet processes for large θ+. This latter result complements large θ results for the Dirichlet process and Poisson–Dirichlet sequences, and complements a recent result on large deviation principles for the two-parameter Poisson–Dirichlet process. A crucial component of our results is the use of distributional identities that may be useful in other contexts.

Chapter information

Source
Bertrand Clarke and Subhashis Ghosal, eds., Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2008), 187-199

Dates
First available in Project Euclid: 28 April 2008

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1209398469

Digital Object Identifier
doi:10.1214/074921708000000147

Mathematical Reviews number (MathSciNet)
MR2459225

Subjects
Primary: 62G05: Estimation
Secondary: 62F15: Bayesian inference

Keywords
Bayesian consistency multiplier CLT two-parameter Poisson–Dirichlet process weak convergence

Rights
Copyright © 2008, Institute of Mathematical Statistics

Citation

James, Lancelot F. Large sample asymptotics for the two-parameter Poisson–Dirichlet process. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 187--199, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2008. doi:10.1214/074921708000000147. https://projecteuclid.org/euclid.imsc/1209398469


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