We introduce a nonparametric regression estimator that is consistent
in the presence of measurement error in the explanatory variable when
one repeated observation of the mismeasured regressor is available. The
approach taken relies on a useful property of the Fourier transform,
namely, its ability to convert complicated integral equations into
simple algebraic equations. The proposed estimator is shown to be
asymptotically normal, and its rate of convergence in probability is
derived as a function of the smoothness of the densities and
conditional expectations involved. The resulting rates are often
comparable to kernel deconvolution estimators, which provide consistent
estimation under the much stronger assumption that the density of the
measurement error is known. The finite-sample properties of the
estimator are investigated through Monte Carlo experiments.This work was made possible in part through
financial support from the National Science Foundation via grant SES-0214068.
The author is grateful to the referees and the co-editor for their helpful
comments.