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May 2022 Local minimax rates for closeness testing of discrete distributions
Joseph Lam-Weil, Alexandra Carpentier, Bharath K. Sriperumbudur
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Bernoulli 28(2): 1179-1197 (May 2022). DOI: 10.3150/21-BEJ1382

Abstract

We consider the closeness testing problem for discrete distributions. The goal is to distinguish whether two samples are drawn from the same unspecified distribution, or whether their respective distributions are separated in L1-norm. In this paper, we focus on adapting the rate to the shape of the underlying distributions, i.e. we consider a local minimax setting. We provide, to the best of our knowledge, the first local minimax rate for the separation distance up to logarithmic factors, together with a test that achieves it. In view of the rate, closeness testing turns out to be substantially harder than the related one-sample testing problem over a wide range of cases.

Funding Statement

The work of A. Carpentier is partially supported by the Deutsche Forschungsgemeinschaft (DFG) Emmy Noether grant MuSyAD (CA 1488/1-1), by the DFG - 314838170, GRK 2297 MathCoRe, by the DFG GRK 2433 DAEDALUS, by the DFG CRC 1294 ‘Data Assimilation’, Project A03, and by the UFA-DFH through the French-German Doktorandenkolleg CDFA 01-18.

Acknowledgements

The authors would like to thank the anonymous referee, the Associate Editor and the Editor for their constructive comments that improved the quality of this paper.

Citation

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Joseph Lam-Weil. Alexandra Carpentier. Bharath K. Sriperumbudur. "Local minimax rates for closeness testing of discrete distributions." Bernoulli 28 (2) 1179 - 1197, May 2022. https://doi.org/10.3150/21-BEJ1382

Information

Received: 1 March 2020; Revised: 1 June 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1382

Keywords: closeness testing , composite-composite testing , discrete distributions , Hypothesis testing , instance optimal , Local minimax optimality , two-sample

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 2 • May 2022
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