Testing independence in high dimensions with sums of squares of rank correlations

We treat the problem of testing independence between $m$ continuous observations when the available sample size $n$ is comparable to $m$. Making no specific distributional assumptions, we consider two related classes of test statistics. Statistics of the first considered type are formed by summing up the squares of all pairwise sample rank correlations. Statistics of the second type are U-statistics that unbiasedly estimate the expected sum of squared rank correlations. In the asymptotic regime where the ratio $m/n$ converges to a positive constant, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits. 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). For degenerate statistics, the asymptotic variance of the test statistics involves a fourth moment of the kernel that does not appear in classical U-statistic theory. The power of the considered tests is explored in rate-optimality theory under a Gaussian equicorrelation alternative as well as in numerical experiments for specific cases of more general alternatives.