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October 2007 On surrogate dimension reduction for measurement error regression: An invariance law
Bing Li, Xiangrong Yin
Ann. Statist. 35(5): 2143-2172 (October 2007). DOI: 10.1214/009053607000000172

Abstract

We consider a general nonlinear regression problem where the predictors contain measurement error. It has been recently discovered that several well-known dimension reduction methods, such as OLS, SIR and pHd, can be performed on the surrogate regression problem to produce consistent estimates for the original regression problem involving the unobserved true predictor. In this paper we establish a general invariance law between the surrogate and the original dimension reduction spaces, which implies that, at least at the population level, the two dimension reduction problems are in fact equivalent. Consequently we can apply all existing dimension reduction methods to measurement error regression problems. The equivalence holds exactly for multivariate normal predictors, and approximately for arbitrary predictors. We also characterize the rate of convergence for the surrogate dimension reduction estimators. Finally, we apply several dimension reduction methods to real and simulated data sets involving measurement error to compare their performances.

Citation

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Bing Li. Xiangrong Yin. "On surrogate dimension reduction for measurement error regression: An invariance law." Ann. Statist. 35 (5) 2143 - 2172, October 2007. https://doi.org/10.1214/009053607000000172

Information

Published: October 2007
First available in Project Euclid: 7 November 2007

zbMATH: 1126.62055
MathSciNet: MR2363966
Digital Object Identifier: 10.1214/009053607000000172

Subjects:
Primary: 62G08 , 62H12

Keywords: central mean space , Central spaces , Invariance , regression graphics , surrogate predictors and response , weak convergence in probability

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 5 • October 2007
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