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$$ \rho $$, Kendall's τ$$ \tau $$, Hoeffding's D, Blum‐Kiefer‐Rosenblatt's R, and Bergsma‐Dassios‐Yanagimoto's τ∗$$ {\tau}^{\ast } $$. The key advantages of our proposed tests are threefold: (1) they do not rely on specific assumptions about the distribution of random vectors, which makes them applicable across a wide range of settings; (2) they can effectively capture nonlinear dependence structures between random vectors, a critical aspect in high‐dimensional contexts; (3) they exhibit robust power performance under both sparse and dense alternatives. Notably, our proposed tests exhibit robust power across a variety of scenarios, as evidenced by extensive numerical results and an empirical application to RNA microarray data.