Dimension reduction regressions with measurement errors subject to additive distortion

In this paper, we propose several dimension reduction methods when the covariates are measured with additive distortion measurement errors. These distortions are modelled by unknown functions of a commonly observable confounding variable. To estimate the central subspace, we propose residuals-based dimension reduction estimation methods and direct estimation methods. The consistency and asymptotic normality of the proposed estimators are investigated. Furthermore, we conduct some simulations to evaluate the performance of our proposed method and compare with existing methods, and a real data set is analysed for illustration.