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 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.