RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS.

To test independence between two high-dimensional random vectors, we propose three tests based on the rank-based indices derived from Hoeffding's $D$, Blum-Kiefer-Rosenblatt's $R$ and Bergsma-Dassios-Yanagimoto's ${\tau }^{*}$. Under the null hypothesis of independence, we show that the distributions of the proposed test statistics converge to normal ones if the dimensions diverge arbitrarily with the sample size. We further derive an explicit rate of convergence. Thanks to the monotone transformation-invariant property, these distribution-free tests can be readily used to generally distributed random vectors including heavily tailed ones. We further study the local power of the proposed tests and compare their relative efficiencies with two classic distance covariance/correlation based tests in high dimensional settings. We establish explicit relationships between $D,R,{\tau }^{*}$ and Pearson's correlation for bivariate normal random variables. The relationships serve as a basis for power comparison. Our theoretical results show that under a Gaussian equicorrelation alternative, (i) the proposed tests are superior to the two classic distance covariance/correlation based tests if the components of random vectors have very different scales; (ii) the asymptotic efficiency of the proposed tests based on $D,{\tau }^{*}$ and $R$ are sorted in a descending order.