Local minimax rates for closeness testing of discrete distributions

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