Stable and bias-corrected estimation for nonparametric regression models

Abstract

It is well known that in nonparametric regression setting, the common kernel estimators are sensitive to bandwidth and can not achieve a satisfactory convergence rate, especially for multivariate cases. To improve nonparametric estimation in the sense of both selection of bandwidth and convergence rate, this paper proposes a two-stage (or three-stage) regression estimation by combining nonparametric regression with parametric regression. The optimal design conditions, including the optimal bandwidth, are obtained. The newly proposed estimator has a simple structure and can achieve a smaller mean square error without use of the higher order kernel. Even if the prior selections of nonparametric estimation are not optimal (i.e. the smooth parameter is not optimally chosen), the new two-stage estimator still has a satisfactory convergence rate. This means that the newly proposed estimator is robust to the selection of bandwidth and then is a practical method. This new method is also suitable for general nonparametric regression models regardless of the dimension of explanatory variable and the structure assumption on regression function.