1Department of Statistics, University of Washington, Box 354322, Seattle, Washington 98195, U.S.A.email@example.comDepartment of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, New York 10027, U.S.A.firstname.lastname@example.orgDepartment of Operations Research and Financial Engineering, Princeton University, Sherrerd Hall, Charlton Street, Princeton, New Jersey 08544, U.S.A.email@example.com
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SummaryWe consider the testing of mutual independence among all entries in a d-dimensional random vector based on n 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 Symbol. 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 d is large.