A Powerful Bootstrap Test of Independence in High Dimensions

Abstract

This paper proposes a nonparametric test of pairwise independence of one random variable from a large pool of other random variables. The test statistic is the maximum of several Chatterjee's rank correlations and critical values are computed via a block multiplier bootstrap. We show in simulations that other popular tests based on distance covariances do not necessarily control size under this null. Our test, on the other hand, is shown to asymptotically control size uniformly over a large class of data-generating processes, even when the number of variables is much larger than sample size. The test is consistent against any fixed alternative. It can be combined with a stepwise procedure for selecting those variables from the pool that violate independence, while controlling the family-wise error rate. All formal results leave the dependence among variables in the pool completely unrestricted. In simulations, we find that our test is typically more powerful than competing methods (in settings where they are valid), particularly in high-dimensional scenarios or when there is dependence among variables in the pool.