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.
翻译:我们考虑离散分布物的近距离测试问题。 目标是区分两个样本是来自同一未指明的分布物,还是它们各自的分布物以1美元- 诺尔值分隔开来。 在本文中,我们侧重于根据基本分布物的形状调整速度, 即我们考虑\ textit{ a local minimax setting} 。 我们根据我们的知识, 提供了离散距离至对数系数的第一个本地微鼠标率, 并同时进行一项达到该比率的测试。 鉴于这一比率, 近距离测试结果比一系列案例的一模数测试问题要困难得多 。