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Dimension reduction for conditional mean in regression. (English) Zbl 1012.62035
Summary: 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. 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.

MSC:
62G08 Nonparametric regression and quantile regression
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62-09 Graphical methods in statistics (MSC2010)
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