Efficient Sparse Estimate of Sufficient Dimension Reduction in High Dimension

In this article, we propose a new efficient sparse estimate (ESE) in sufficient dimension reduction using distance covariance. Our method is model-free and does not need any kernel function or slicing selection. Moreover, it can naturally deal with multivariate response scenarios, making it appealing in a modified sequential algorithm that targets the large <i>p</i> small <i>n</i> problems. Compared with screening procedures that only use marginal utility, our method can extract more useful information from the data and is capable of determining the size of the selected submodel automatically while most of screening procedures cannot. Under mild conditions, based on manifold theories and techniques, it can be shown that our method would perform asymptotically as well as if the true irrelevant predictors were known, which is referred to as the oracle property. Extensive simulation studies and two real data examples demonstrate the effectiveness and efficiency of the proposed approach. It is remarkable that the analysis in cardiomyopathy microarray data reveals distinct and interesting findings. Supplemental materials for this article are available online.