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Projection correlation between scalar and vector variables and its use in feature screening with multi-response data

In this article, we introduce a new methodology to perform feature screening for ultrahigh dimensional data with multivariate responses. Several extant screening procedures are available for multivariate responses, but they may be adversely affected by heavy-tailed observations or the dimension of multivariate responses. In order to attack these challenges, we first introduce a nonparametric coefficient, called projection correlation, to measure the departure of dependence between a scalar variable X and a vector variable . It takes values between zero and one, does not require any moment conditions on X and , and is zero if and only if X and are independent. Based on its estimation that has desirable theoretical properties, such as algebraic simplicity and consistency, we present a novel sure independence screening procedure, which enjoys the desirable sure screening property. Numerical results demonstrate the effectiveness of the proposed procedure in comparison with the existing counterparts.