Title: A Berry-Esseen theorem for Pitman's alpha-diversity
Authors: Stefano Favaro - University of Torino and Collegio Carlo Alberto (Italy) [presenting]
Abstract: The purpose is to study the random number $K_n$ of blocks in the exchangeable random partition induced by a random sample of size $n$ from the two parameter Poisson-Dirichlet process prior. Our main result is a Berry-Esseen theorem for the large $n$ asymptotic behaviour of $K_n$. The proof relies on three intermediate novel results which may be of independent interest: i) a (probabilistic) representation of the distribution of $K_n$ in terms of a compound Poisson distribution; ii) a quantitative version of the classical asymptotic expansion, in the sense of Poincar\'e, of a recurrent Laplace-type integral; iii) a refined quantitative bound for classical Poisson approximation. An application of our Berry-Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying explicitly the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.