TESTING INDEPENDENCE IN HIGH DIMENSIONS WITH SUMS OF RANK CORRELATIONS

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 domin..