The Annals of Statistics

Dimension reduction for conditional mean in regression

R.Dennis Cook and Bing Li

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In many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors, and less concerned with the other aspects of the conditional distribution. In this paper we develop dimension reduction methods that incorporate this consideration. We introduce the notion of the Central Mean Subspace (CMS), a natural inferential object for dimension reduction when the mean function is of interest. We study properties of the CMS, and develop methods to estimate it. These methods include a new class of estimators which requires fewer conditions than pHd, and which displays a clear advantage when one of the conditions for pHd is violated. CMS also reveals a transparent distinction among the existing methods for dimension reduction: OLS, pHd, SIR and SAVE. We apply the new methods to a data set involving recumbent cows.

Article information

Ann. Statist., Volume 30, Number 2 (2002), 455-474.

First available in Project Euclid: 14 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62-09: Graphical methods 62H05: Characterization and structure theory

Central subspace graphics regression pHd SAVE SIR visualization


Cook, R.Dennis; Li, Bing. Dimension reduction for conditional mean in regression. Ann. Statist. 30 (2002), no. 2, 455--474. doi:10.1214/aos/1021379861.

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