Testing independence in high dimensions with sums of rank correlations

TL;DR

This paper develops and analyzes high-dimensional independence tests based on sums of rank correlations, establishing their asymptotic Gaussian limits without distributional assumptions, and evaluates their power under various alternatives.

Contribution

It introduces a framework for testing independence in high dimensions using sums of rank correlations with proven asymptotic Gaussian distributions, covering a wide class of rank-based U-statistics.

Findings

Asymptotic null distributions are Gaussian for large m and n.

The tests are rate-optimal under Gaussian equicorrelation alternatives.

Numerical experiments confirm the effectiveness of the proposed tests.

Abstract

We treat the problem of testing independence between m continuous variables when m can be larger than the available sample size n. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both m and n tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no specific distributional or moment assumptions on the data. Using the framework of U-statistics, our result covers a variety of rank correlations including Kendall's tau and a dominating term of Spearman's rank correlation coefficient (rho), but also degenerate U-statistics such as Hoeffding's $D$, or the $\tau^*$ of Bergsma and Dassios (2014). As in the classical theory for U-statistics, the test statistics need to be scaled differently when the rank correlations used to construct them are degenerate U-statistics. The power of the considered tests is explored in rate-optimality theory under Gaussian equicorrelation alternatives as well as in numerical experiments for specific cases of more general alternatives.