Title: Moderate Deviations for Ewens-Pitman Sampling Models

Author(s): Stefano Favaro, Shui Feng and Fuqing Gao
Issue: Volume 80 Series A Part 2 Year 2018
Pages: 330 -- 341
Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the Poisson-Dirichlet distribution with parameter $\alpha$ \in [0, 1)$ and $\theta > − $\alpha$. Given a sample of size $n$ from the population, two important statistics are the number $K_n$ of different types in the sample, and the number $M_{l, n}$ of different types with frequency $l$ in the sample. We establish moderate deviation principles for $(K_n)_{n \ge 1}$ and $(M_{l, n})_{n \ge 1}$. Corresponding rate functions are explicitly identified, which help in revealing a critical scale and in understanding the exact role of the parameters $\alpha$ and $\theta$.
AMS (2000) subject classification. Primary 60F05; Secondary 60G57.
Keywords and phrases: $\alpha$-diversity, Large and moderate deviations, Poisson- Dirichlet distribution, Random partition.