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

TL;DR

This paper establishes an invariance law linking surrogate and original dimension reduction spaces in measurement error regression, enabling the use of existing methods for consistent estimation, with theoretical and empirical validation.

Contribution

It introduces a general invariance law that shows the equivalence of surrogate and original dimension reduction spaces, facilitating the application of existing methods in measurement error contexts.

Findings

The invariance law holds exactly for multivariate normal predictors.

The law approximately applies to arbitrary predictors.

Surrogate dimension reduction estimators converge at a quantifiable rate.

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.