## The Annals of Statistics

### On surrogate dimension reduction for measurement error regression: An invariance law

#### 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.

#### Article information

Source
Ann. Statist., Volume 35, Number 5 (2007), 2143-2172.

Dates
First available in Project Euclid: 7 November 2007

https://projecteuclid.org/euclid.aos/1194461725

Digital Object Identifier
doi:10.1214/009053607000000172

Mathematical Reviews number (MathSciNet)
MR2363966

Zentralblatt MATH identifier
1126.62055

Subjects
Primary: 62G08: Nonparametric regression 62H12: Estimation

#### Citation

Li, Bing; Yin, Xiangrong. On surrogate dimension reduction for measurement error regression: An invariance law. Ann. Statist. 35 (2007), no. 5, 2143--2172. doi:10.1214/009053607000000172. https://projecteuclid.org/euclid.aos/1194461725

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