Empirical and Full Bayes estimation of the type of a Pitman-Yor process

The Pitman-Yor process is a random discrete probability distribution of which the atoms can be used to model the relative abundance of species. The process is indexed by a type parameter \(\sigma\), which controls the number of different species in a finite sample from a realization of the distribut...

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Bibliographic Details
Published in:arXiv.org 2022-08
Main Authors: S E M P Franssen, A W van der Vaart
Format: Article
Language:English
Subjects:
Asymptotic methods
Likelihood ratio
Normality
Process parameters
Online Access:Get full text
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Summary:The Pitman-Yor process is a random discrete probability distribution of which the atoms can be used to model the relative abundance of species. The process is indexed by a type parameter \(\sigma\), which controls the number of different species in a finite sample from a realization of the distribution. A random sample of size \(n\) from the Pitman-Yor process of type \(\sigma>0\) will contain of the order \(n^\sigma\) distinct values (``species''). In this paper we consider the estimation of the type parameter by both empirical Bayes and full Bayes methods. We derive the asymptotic normality of the empirical Bayes estimator and a Bernstein-von Mises theorem for the full Bayes posterior, in the frequentist setup that the observations are a random sample from a given true distribution. We also consider the estimation of the second parameter of the Pitman-Yor process, the prior precision. We apply our results to derive the limit behaviour of the likelihood ratio in a setting of forensic statistics.
ISSN:2331-8422