Kernel‐based marginal testing for covariate effects in high‐dimensional settings

Abstract In high dimensions, the relationship between covariates and a response variable becomes increasingly intricate, with different covariate components often displaying varying degrees of variability. This complex interplay of dependence, heterogeneity, and high dimensionality presents a significant challenge when investigating the effects of covariates on the response variable. To address this, we propose a novel marginal testing procedure based on kernel‐based conditional mean dependence, which can be implemented without requiring model assumptions. Theoretically, we establish the limiting normal distributions of the test statistic under both null hypotheses and local alternatives by asymptotically approximating a class of quadratic forms. We also examine the asymptotic relative efficiency of the proposed test against several state‐of‐the‐art alternatives. Our theoretical evaluations are conducted from two perspectives: a detailed analysis within a linear model framework and a comparison within a fully non‐parametric setting. The effectiveness and applicability of the proposed method are demonstrated through both simulation studies and real data analysis.