Some Stabilized Bandwidth Selectors for Nonparametric Regression

Abstract

The problem of bandwidth selection for nonparametric kernel regression is considered. It is well recognized that the classical bandwidth selectors are subject to large sample variation. Due to the large variation, these selectors might not be very useful in practice. Based on the frequency-domain representation of the residual sum of squares (RSS), the source of the variation is pointed out. The observation leads to consideration of a procedure which stabilizes the RSS by modifying the periodogram of the observations. The stabilized bandwidth selectors are obtained by substituting the stabilized RSS for the RSS in the classical selectors. The strong consistency of the stabilized bandwidth estimate is established. For sufficiently smooth regression functions, it is shown that the stabilized bandwidth is asymptotically normal, and the relative convergence rate of the stabilized bandwidth estimate is $T^{-1/2}$ instead of the rate $T^{-1/10}$ of the classical estimates. In a simulation study, it is confirmed that the stabilized selectors perform much better than the classical selectors. The simulation results are consistent with the theoretic results. The article contains the important message that the commonly used cross-validation can be improved substantially. The procedure and the theoretic results are developed for a rather restrictive case. Further studies are required for more general situations.