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February 2013 A general theory for nonlinear sufficient dimension reduction: Formulation and estimation
Kuang-Yao Lee, Bing Li, Francesca Chiaromonte
Ann. Statist. 41(1): 221-249 (February 2013). DOI: 10.1214/12-AOS1071

Abstract

In this paper we introduce a general theory for nonlinear sufficient dimension reduction, and explore its ramifications and scope. This theory subsumes recent work employing reproducing kernel Hilbert spaces, and reveals many parallels between linear and nonlinear sufficient dimension reduction. Using these parallels we analyze the properties of existing methods and develop new ones. We begin by characterizing dimension reduction at the general level of $\sigma$-fields and proceed to that of classes of functions, leading to the notions of sufficient, complete and central dimension reduction classes. We show that, when it exists, the complete and sufficient class coincides with the central class, and can be unbiasedly and exhaustively estimated by a generalized sliced inverse regression estimator (GSIR). When completeness does not hold, this estimator captures only part of the central class. However, in these cases we show that a generalized sliced average variance estimator (GSAVE) can capture a larger portion of the class. Both estimators require no numerical optimization because they can be computed by spectral decomposition of linear operators. Finally, we compare our estimators with existing methods by simulation and on actual data sets.

Citation

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Kuang-Yao Lee. Bing Li. Francesca Chiaromonte. "A general theory for nonlinear sufficient dimension reduction: Formulation and estimation." Ann. Statist. 41 (1) 221 - 249, February 2013. https://doi.org/10.1214/12-AOS1071

Information

Published: February 2013
First available in Project Euclid: 26 March 2013

zbMATH: 1347.62018
MathSciNet: MR3059416
Digital Object Identifier: 10.1214/12-AOS1071

Subjects:
Primary: 62B05, 62G08, 62H30

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 1 • February 2013
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