Testing mutual independence in high dimension via distance covariance

We introduce an L2‐type test for testing mutual independence and banded dependence structure for high dimensional data. The test is constructed on the basis of the pairwise distance covariance and it accounts for the non‐linear and non‐monotone dependences among the data, which cannot be fully captured by the existing tests based on either Pearson correlation or rank correlation. Our test can be conveniently implemented in practice as the limiting null distribution of the test statistic is shown to be standard normal. It exhibits excellent finite sample performance in our simulation studies even when the sample size is small albeit the dimension is high and is shown to identify non‐linear dependence in empirical data analysis successfully. On the theory side, asymptotic normality of our test statistic is shown under quite mild moment assumptions and with little restriction on the growth rate of the dimension as a function of sample size. As a demonstration of good power properties for our distance‐covariance‐based test, we further show that an infeasible version of our test statistic has the rate optimality in the class of Gaussian distributions with equal correlation.