Testing Independence Between High-Dimensional Random Vectors Using Rank-Based Max-Sum Tests

In this paper, we address the problem of testing independence between two high-dimensional random vectors. Our approach involves a series of max-sum tests based on three well-known classes of rank-based correlations. These correlation classes encompass several popular rank measures, including Spearman's $ρ$, Kendall's $τ$, Hoeffding's D, Blum-Kiefer-Rosenblatt's R and Bergsma-Dassios-Yanagimoto's $τ^*$.The key advantages of our proposed tests are threefold: (1) they do not rely on specific assumptions about the distribution of random vectors, which flexibility makes them available across various scenarios; (2) they can proficiently manage non-linear dependencies between random vectors, a critical aspect in high-dimensional contexts; (3) they have robust performance, regardless of whether the alternative hypothesis is sparse or dense.Notably, our proposed tests demonstrate significant advantages in various scenarios, which is suggested by extensive numerical results and an empirical application in RNA microarray analysis.