Dimension Reduction in Regressions through Weighted Variance Estimation

Abstract

Because sliced inverse regression (SIR) using the conditional mean of the inverse regression fails to recover the central subspace when the inverse regression mean degenerates, sliced average variance estimation (SAVE) using the conditional variance was proposed in the sufficient dimension reduction literature. However, the efficacy of SAVE depends heavily upon the number of slices. In the present article, we introduce a class of weighted variance estimation (WVE), which, similar to SAVE and simple contour regression (SCR), uses the conditional variance of the inverse regression to recover the central subspace. The strong consistency and the asymptotic normality of the kernel estimation of WVE are established under mild regularity conditions. Finite sample studies are carried out for comparison with existing methods and an application to a real data is presented for illustration.

Keywords:

• Asymptotic normality
• Dimension reduction
• Inverse regression
• Simple contour regression
• Sliced average variance estimation

Mathematics Subject Classification:

• 62J05
• 62J07
• 62F35
• 62G35

Acknowledgments

The first author was supported by a NSF grant from National Natural Science Foundation of China (No. 10701035), and ChenGuang project of Shanghai Education Development Foundation (No. 2007CG33), and the second author was supported by a grant (HKU 7058/05P) from the Research Grants Council of Hong Kong, and a FRG grant from Hong Kong Baptist University, Hong Kong.