  We 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 propose tests that control the type I error in the
high-dimensional setting where $d>n$. We further show that the two tests are
rate-optimal in terms of power against sparse alternatives, and outperform
competitors in simulations, especially when $d$ is large.
