A Note on Multiple Regression for Single Index Model

Abstract

A simple method based on sliced inverse regression (SIR) is proposed to explore an effective dimension reduction (EDR) vector for the single index model. We avoid the principle component analysis step of the original SIR by using two sample mean vectors in two slices of the response variable and their difference vector. The theories become simpler, the method is equivalent to the multiple linear regression with dichotomized response, and the estimator can be expressed by a closed form, although the objective function might be an unknown nonlinear. It can be applied for the case when the number of covariates is large, and it requires no matrix operation or iterative calculation.

Key Words:

• Correlation coefficient
• Effective dimension reduction
• Semiparametric regression
• Single index model
• Sliced inverse regression

Mathematics Subject Classification:

• Primary 62G08
• Secondary 62J05

Acknowledgments

The authors thank Dr. J. B. Cologne for his helpful comments and the referees for their encouragement and valuable advice. This study was supported in part by Grant-in-Aid for Encouragement of Young Scientists from the Ministry of Education Science, Sport Culture of Japan.