Design-adaptive Local Polynomial Estimator for the Errors-in-Variables Problem

Local polynomial estimators are popular techniques for nonparametric regression estimation and have received great attention in the literature. Their simplest version, the local constant estimator, can be easily extended to the errors-in-variables context by exploiting its similarity with the deconvolution kernel density estimator. The generalization of the higher order versions of the estimator, however, is not straightforward and has remained an open problem for the last 15 years, since the publication of Fan and Truong (1993). We propose an innovative local polynomial estimator of any order in the errors-in-variables context, derive its design-adaptive asymptotic properties and study its finite sample performance on simulated examples. This provides not only a solution to this outstanding problem, but also methodological contributions to error-in-variable regression, including local polynomial estimation of derivative functions.