Martingale Difference Correlation and Its Use in High-Dimensional Variable Screening

In this article, we propose a new metric, the so-called martingale difference correlation, to measure the departure of conditional mean independence between a scalar response variable V and a vector predictor variable U. Our metric is a natural extension of distance correlation proposed by Székely, Rizzo, and Bahirov, which is used to measure the dependence between V and U. The martingale difference correlation and its empirical counterpart inherit a number of desirable features of distance correlation and sample distance correlation, such as algebraic simplicity and elegant theoretical properties. We further use martingale difference correlation as a marginal utility to do high-dimensional variable screening to screen out variables that do not contribute to conditional mean of the response given the covariates. Further extension to conditional quantile screening is also described in detail and sure screening properties are rigorously justified. Both simulation results and real data illustrations demonstrate the effectiveness of martingale difference correlation-based screening procedures in comparison with the existing counterparts. Supplementary materials for this article are available online.