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On regression estimators with de-noised variables. (English) Zbl 1004.62038
Summary: We consider linear measurement error models where the variables in error are observed together with an auxiliary variable, say, time. Z. Cai, P.A. Naik and C.-L. Tsai [Stat. Sin. 10, No. 4, 1231-1241 (2000; Zbl 0960.62134)] studied this problem and proposed using a de-noising process prior to a least squares analysis. The present paper focuses on the asymptotic distributions of such de-noised estimators. We demonstrate that the use of de-noising contributes to an efficiency gain over other estimators of measurement error models that do not make use of any auxiliary information. We also extend the results to cases with dependent errors, and to a general class of $$M$$-estimators that have better robustness properties than least squares.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62J05 Linear regression; mixed models 62E20 Asymptotic distribution theory in statistics