Extending the Scope of Inverse Regression Methods in Sufficient Dimension Reduction

Abstract

In the area of sufficient dimension reduction, two structural conditions are often assumed: the linearity condition that is close to assuming ellipticity of underlying distribution of predictors, and the constant variance condition that nears multivariate normality assumption of predictors. Imposing these conditions are considered as necessary trade-off for overcoming the “curse of dimensionality”. However, it is very hard to check whether these conditions hold or not. When these conditions are violated, some methods such as marginal transformation and re-weighting are suggested so that data fulfill them approximately. In this article, we assume an independence condition between the projected predictors and their orthogonal complements which can ensure the commonly used inverse regression methods to identify the central subspace of interest. The independence condition can be checked by the gridded chi-square test. Thus, we extend the scope of many inverse regression methods and broaden their applicability in the literature. Simulation studies and an application to the car price data are presented for illustration.

Keywords:

• Ellipticity
• Inverse regression
• Linearity condition
• Sliced inverse regression
• Sufficient dimension reduction

Mathematics Subject Classification:

• 62G09
• 62H12

Acknowledgment

This work was supported by a NSF grant from National Natural Science Foundation of China (No. 10701035), ChenGuang project of Shanghai Education Development Foundation (No. 2007CG33).