Successive direction extraction for estimating the central subspace in a multiple-index regression

In this paper we propose a dimension reduction method to estimate the directions in a multiple-index regression based on information extraction. This extends the recent work of Yin and Cook (2005) who introduced the method and used it to estimate the direction in a single-index regression. While a formal extension seems conceptually straightforward, there is a fundamentally new aspect of our extension: We are able to show that, under the assumption of elliptical predictors, the estimation of multiple index regressions can be decomposed into successive single-index estimation problems. This significantly reduces the computational complexity, because the nonparametric procedure involves only a one-dimensional search at each stage. In addition, we developed a permutation test to assist in estimating the dimension of a multiple-index regression.