Distribution-Free Tests of Independence in High Dimensions

TL;DR

This paper introduces distribution-free tests for mutual independence in high-dimensional data, demonstrating their asymptotic behavior, optimal power against sparse alternatives, and superior performance in simulations.

Contribution

It develops new high-dimensional independence tests based on Kendall's tau and Spearman's rho, with proven asymptotic properties and improved power over existing methods.

Findings

Test statistics converge to Gumbel distributions under null hypothesis.

Proposed tests control type I error in high-dimensional settings.

Tests are rate-optimal and outperform competitors in simulations.

Abstract

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.