Nearly minimax optimal Wasserstein conditional independence testing

Abstract This paper is concerned with minimax conditional independence testing. In contrast to some previous works on the topic, which use the total variation distance to separate the null from the alternative, here we use the Wasserstein distance. In addition, we impose Wasserstein smoothness conditions that on bounded domains are weaker than the corresponding total variation smoothness imposed, for instance, by Neykov et al. (2021, Ann. Statist., 49, 2151–2177). This added flexibility expands the distributions that are allowed under the null and the alternative to include distributions that may contain point masses for instance. We characterize the optimal rate of the critical radius of testing up to logarithmic factors. Our test statistic that nearly achieves the optimal critical radius is novel, and can be thought of as a weighted multi-resolution version of the $U$-statistic studied by Neykov et al. (2021, Ann. Statist., 49, 2151–2177).