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

Abstract

In this paper we propose a dimension reduction method for estimating the directions in a multiple-index regression based on information extraction. This extends the recent work of Yin and Cook [X. Yin, R.D. Cook, Direction estimation in single-index regression, Biometrika 92 (2005) 371-384] 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.

The Dual Central Subspaces in dimension reduction

“…Next, a direct extension of the proof of proposition 1 in Yin & Cook (2008) to coefficient matrices gives,…”

A theoretical note on optimal sufficient dimension reduction with singularity

“…Also, define S Y|X as the central subspace, which is the intersection among all possible dimension reduction subspaces. To guarantee the existence of S Y|X , several conditions are required, which are well described in Cook (1998) and Yin et al (2008). In SDR literature, the estimation of S Y|X is of primary interest.…”

Using sliced inverse mean difference for sufficient dimension reduction

“…The ultimate goal is to estimate accurately the matrix β whose column space spans the CS. Conditions of existence of the CS are given in Cook (1998a) and Yin et al (2008). These conditions are mild and throughout this paper we assume the existence of CS.…”