A Statistically and Numerically Efficient Independence Test Based on Random Projections and Distance Covariance

Abstract

Testing for independence plays a fundamental role in many statistical techniques. Among the nonparametric approaches, the distance-based methods (such as the distance correlation-based hypotheses testing for independence) have many advantages, compared with many other alternatives. A known limitation of the distance-based method is that its computational complexity can be high. In general, when the sample size is n, the order of computational complexity of a distance-based method, which typically requires computing of all pairwise distances, can be O(n2). Recent advances have discovered that in the univariate cases, a fast method with O(n log n) computational complexity and O(n) memory requirement exists. In this paper, we introduce a test of independence method based on random projection and distance correlation, which achieves nearly the same power as the state-of-the-art distance-based approach, works in the multivariate cases, and enjoys the O(nK log n) computational complexity and O(max{n, K}) memory requirement, where K is the number of random projections. Note that saving is achieved when K