We consider a novel multivariate nonparametric two-sample testing problem where, under the alternative, distributions $P$ and $Q$ are separated in an integral probability metric over functions of bounded total variation (TV IPM). We propose a new test, the graph TV test, which uses a graph-based approximation to the TV IPM as its test statistic. We show that this test, computed with an $\varepsilon$-neighborhood graph and calibrated by permutation, is minimax rate-optimal for detecting alternatives separated in the TV IPM. As an important special case, we show that this implies the graph TV test is optimal for detecting spatially localized alternatives, whereas the $\chi^2$ test is provably suboptimal. Our theory is supported with numerical experiments on simulated and real data.

翻译：我们研究一种新颖的多元非参数双样本检验问题，其中在备择假设下，分布$P$和$Q$在有界总变差函数空间上的积分概率度量（TV IPM）中具有分离性。我们提出了一种新的检验方法——图TV检验，该方法使用基于图的总变差积分概率度量近似作为其检验统计量。我们证明，采用$\varepsilon$邻域图计算并通过置换校准的该检验，在检测TV IPM分离的备择假设时达到极小极大速率最优性。作为一个重要的特例，我们证明这意味着图TV检验对于检测空间局部化备择假设是最优的，而$\chi^2$检验被证明是次优的。我们的理论得到了在模拟数据和真实数据上的数值实验支持。