Distribution-Free Tests of Independence in High Dimensions

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.Comment: to appear in Biometrik

Scale Invariant Conditional Dependence Measures

<p>In this paper we develop new dependence and conditional dependence measures and provide their estimators. An attractive property of these measures and estimators is that they are invariant to any monotone increasing transformations of the random variables, which is important in many applications including feature selection. Under certain conditions we show the consistency of these estimators, derive upper bounds on their convergence rates, and show that the estimators do not suffer from the curse of dimensionality. However, when the conditions are less restrictive, we derive a lower bound which proves that in the worst case the convergence can be arbitrarily slow similarly to some other estimators. Numerical illustrations demonstrate the applicability of our method.</p