Moderate deviations for Ewens-Pitman exchangeable random partitions

TL;DR

This paper establishes moderate deviation principles for key statistics of samples from populations with Poisson-Dirichlet distributed types, providing explicit rate functions and insights into parameter roles.

Contribution

It introduces the first moderate deviation principles for the total number of types and their frequencies in Poisson-Dirichlet models, including posterior versions.

Findings

Explicit rate functions for deviations are derived.

Critical scales for deviations are identified.

Posterior deviation principles are established.

Abstract

Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, or posterior, counterparts of moderate deviation principles are also established.