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.
Bibliographical noteFunding Information:
The second author’s work is supported in part by National Science Foundation grants DMS-0204662 and DMS-0405681. He would like to thank Anand Vidyashankar for support for visiting UGA where part of this work was done.
The third author’s work was supported in part by National Science Foundation grants DMS-0405360 and 0704098.
- Dimension reduction subspaces
- Permutation test
- Regression graphics
- Sufficient dimension reduction