High-dimensional Variable Screening via Conditional Martingale Difference Divergence

Variable screening has been a useful research area that helps to deal with ultra-high-dimensional data. When there exist both marginally and jointly dependent predictors to the response, existing methods such as conditional screening or iterative screening often suffer from the instability against the selection of the conditional set or the computational burden, respectively. In this paper, we propose a new independence measure, named conditional martingale difference divergence (CMDH), that can be treated as either a conditional or a marginal independence measure. Under regularity conditions, we show that the sure screening property of CMDH holds for both marginally and jointly active variables. Based on this measure, we propose a kernel-based model-free variable screening method that is efficient, flexible, and stable against high correlation and heterogeneity. In addition, we provide a data-driven method of conditional set selection, when the conditional set is unknown. In simulations and real data applications, we demonstrate the superior performance of the proposed method.