Variation of conditional mean and its application in ultrahigh dimensional feature screening

A new metric, called variation of conditional mean (VCM), is proposed to measure the dependence of conditional mean of a response variable on a predictor variable. The VCM has several appealing merits. It equals zero if and only if the conditional mean of the response is independent of the predictor; it can be used for both real vector-valued variables and functional data. An estimator of the VCM is given through kernel smoothing, and a test for the conditional mean independence based on the estimated VCM is constructed. The limit distributions of the test statistic under the null hypothesis and alternative hypothesis are deduced, respectively. We further use VCM as a marginal utility to do high-dimensional feature screening to screen out variables that do not contribute to the conditional mean of the response given the predictors and prove the validity of the sure screening property. Furthermore, we find the cross variation of conditional mean (CVCM), a variant of the VCM, has a faster convergence rate than the VCM under conditional mean independence. Numerical comparison shows that the VCM and CVCM performs well in both conditional independence testing and feature screening. We also illustrate their applications to real data sets.