Abstract
Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the Poisson-Dirichlet distribution with parameter α ∈ [0,1) and 𝜃 > −α. Given a sample of size n from the population, two important statistics are the number Kn of different types in the sample, and the number Ml,n of different types with frequency l in the sample. We establish moderate deviation principles for (Kn)n≥ 1 and (Ml,n)n≥ 1. Corresponding rate functions are explicitly identified, which help in revealing a critical scale and in understanding the exact role of the parameters α and 𝜃.
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Acknowledgements
The authors are grateful to two anonymous Referees for valuable remarks. The authors acknowledge the Banff International Research Station (BIRS), Canada, where this project has been completed during the Research in Team Programme “Random partitions and Bayesian nonparametrics”. S. Favaro is supported by the European Research Council through StG N-BNP 306406. Shui Feng is supported by the Natural Sciences and Engineering Research Council of Canada.
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Favaro, S., Feng, S. & Gao, F. Moderate Deviations for Ewens-Pitman Sampling Models. Sankhya A 80, 330–341 (2018). https://doi.org/10.1007/s13171-018-0124-z
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DOI: https://doi.org/10.1007/s13171-018-0124-z