Nonparametric Regression with Errors in Variables

The Annals of Statistics · 1993 · Volume 21 · Issue 4

Abstract

The effect of errors in variables in nonparametric regression estimation is examined. To account for errors in covariates, deconvolution is involved in the construction of a new class of kernel estimators. It is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution. In fact, there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth. It is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate, which is rich enough for many practical applications. Furthermore, to achieve optimality, we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators.

Keywords

62G05; 62G20; 62J99; Deconvolution; errors in variables; Kernel estimator; Nonparametric regression; Optimal rates of convergence;

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