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 -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.
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.
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.
"Local minimax rates for closeness testing of discrete distributions." Bernoulli 28 (2) 1179 - 1197, May 2022. https://doi.org/10.3150/21-BEJ1382