On a Projective Resampling Method for Dimension Reduction With Multivariate Responses

Abstract

Consider the dimension reduction problem where both the response and the predictor are vectors. Existing estimators of this problem take one of the following routes: (1) targeting the part of the dimension reduction space that is related to the conditional mean (or moments) of the response vector, (2) pooling the estimates for the marginal dimension reduction spaces, and (3) estimating the whole dimension reduction space directly by multivariate slicing. However, the first two approaches do not fully recover the dimension reduction space, and the third is hampered by the fact that the accuracy of estimators based on multivariate slicing drops sharply as the dimension of response increases—a phenomenon often called the “curse of dimensionality.” We propose a new method that overcomes both difficulties, in that it involves univariate slicing only and it is guaranteed to fully recover the dimension reduction space under reasonable conditions. The method will be compared with the existing estimators by simulation and applied to a dataset.