Dimension reduction via an alternating inverse regression

In this article, we propose an alternating inverse regression (AIR) to estimate the central subspace (CS) in a successive manner. Taking advantage of both sliced inverse regression (SIR) and partial least squares (PLS), AIR circumvents the collinearity and curse of dimensionality simultaneously. A modified BIC criterion with a penalty term achieving an optimal convergence rate is suggested to estimate the dimension of the CS. We also extend AIR to the multivariate responses case. Through illustrative examples and a real dataset, we demonstrate the usefulness of AIR, and its advantages over some existing methods. Supplemental materials for this article are available online.