Normal Approximation Rate and Bias Reduction for Data-Driven Kernel Smoothing Estimator in a Semiparametric Regression Model

Accuracy of the normal approximation for Speckmanʹs kernel smoothing estimator of the parametric component β in the semiparametric regression model y=xτβ+g(t)+e is studied when the bandwidth used in the estimator is selected by a general data-based method which includes such commonly used bandwidth selectors as (delete-one-out) CV, GCV, and Mallowsʹ CL criterion. We find that, contrary to what we might expect, this data-driven estimator cannot attain the optimal Berry–Esseen rate n−1/2. Consequently, the confidence region of β based on this normal approximation is not first-order accurate. The reason for this is that the bias of Speckmanʹs estimator is still of nonparametric order at the data-driven bandwidth choice. We then propose a resmoothing method to reduce the bias and show that the proposed estimator can achieve the optimal Berry–Esseen rate. A simulation study shows a slightly better small-sample performance of the proposed estimator.