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

• 成果类型: Article
• 署名作者: Li, Bing; Yin, Xiangrong
• 署名单位: Pennsylvania Commonwealth System of Higher Education (PCSHE); Pennsylvania State University; Pennsylvania State University - University Park; University System of Georgia; University of Georgia
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053607000000172
• 发表日期: 2007
• 页码: 2143-2172
• 关键词: principal hessian directions; Sliced Inverse Regression; data visualization; link; moments; models

摘要

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.