Global adaptive smoothing regression

Abstract

We propose an adaptive smoothing method for nonparametric regression. The central idea of the proposed method is to “calibrate” the estimated function through an adaptive bandwidth function, which is a kind of intermediate solution between the global bandwidth (constant on the support) and the local bandwidth (variable with $x$). This also allows to correct the bias of the local polynomial estimator, with some benefits for the inference based on such estimators. Our method, which uses the Neural Network technique in a preliminary (pilot) stage, is based on a rolling, plug-in, bandwidth selection procedure. It automatically reaches a trade-off between the efficiency of global smoothing and the adaptability of local smoothing. The consistency and the optimal convergence rate of the resulting bandwidth estimators are shown theoretically. A simulation study shows the performance of our method for finite sample size.