# Local minimax rates for closeness testing of discrete distributions

## TL;DR

This paper establishes the first local minimax rates for closeness testing of discrete distributions, adapting to the distribution shape, and introduces a test that achieves these rates, highlighting increased difficulty over one-sample testing.

## Contribution

It provides the first local minimax rate for closeness testing and proposes a test that attains this rate, considering the distribution shape.

## Key findings

- First local minimax rate established for closeness testing.
- Proposed test achieves the derived minimax rate.
- Closeness testing is more challenging than one-sample testing in many cases.

## 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 $L_1$-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.