Powerful kernel‐based association tests for multivariate responses

In modern statistics, testing for dependence is often essential. The complex relationships between covariates and response variables present significant analytical challenges. While kernel‐based independence analysis has become a powerful alternative to tackle these issues, there is currently no universally effective kernel‐based test available. We introduce a series of kernel‐based independence tests within the framework of reproducing kernel Hilbert spaces (RKHS). This work includes explicit sample‐level expressions for these tests, as well as their asymptotic null distributions. Additionally, we develop two specific tests: The Maximal Kernel‐based Independence Test (MKIT) and the Maximin Efficient Robust Test (MERT), both of which are derived from the broader category of kernel‐based independence tests. Theoretically, we prove that MKIT and MERT asymptotically conform to the extreme‐value type I‐Gumbel distribution and the normal distribution under certain regular conditions, respectively, and analyze the powers of MKIT and MERT. We conduct extensive simulations that show that MKIT and MERT outperform numerous existing methods across various scenarios. Applications to heterogeneous stock mice data and human connectome project data further highlight the superior performance of the proposed test methods.