Composite Coefficient of Determination and Its Application in Ultrahigh Dimensional Variable Screening

In this article, we propose to measure the dependence between two random variables through a composite coefficient of determination (CCD) of a set of nonparametric regressions. These regressions take consecutive binarizations of one variable as the response and the other variable as the predictor. The resulting measure is invariant to monotonic marginal variable transformation, rendering it robust against heavy-tailed distributions and outliers, and convenient for independent testing. Estimation of CCD could be done through kernel smoothing, with a consistency rate of root-n. CCD is a natural measure of the importance of variables in regression and its sure screening property, when used for variable screening, is also established. Comprehensive simulation studies and real data analysis show that the newly proposed measure quite often turns out to be the most preferred compared to other existing methods both in independence testing and in variable screening. Supplementary materials for this article are available online.