Local minimax rates for closeness testing of discrete distributions
This work addresses a fundamental statistical problem for researchers in distribution testing, offering a local minimax analysis that adapts to distribution shapes, but it is incremental as it builds on prior testing frameworks.
The paper tackles the closeness testing problem for discrete distributions by distinguishing whether two samples come from the same distribution or are separated in L1-norm, and it provides the first local minimax rate for the separation distance up to logarithmic factors, showing that closeness testing is substantially harder than one-sample testing in many cases.
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 $L_1$-norm. In this paper, we focus on adapting the rate to the shape of the underlying distributions, i.e. we consider \textit{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.