Empirical and Full Bayes estimation of the type of a Pitman-Yor process
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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 σ, 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 σ>0 will contain of the order n^σ 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.
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