Distribution-free tests of independence in high dimensions

We consider the testing of mutual independence among all entries in a [Formula: see text]-dimensional random vector based on [Formula: see text] independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and we propose tests that control the Type I error in the high-dimensional setting where [Formula: see text]. We further show that the two tests are rate-optimal in terms of power against sparse alternatives and that they outperform competitors in simulations, especially when [Formula: see text] is large.