Local Bandwidth Selection for Kernel Estimates

Abstract

Abstract A kernel estimate of a curve that uses an adaptive procedure for local selection of the bandwidth is considered here. A two-step procedure is proposed for estimating the local bandwidth that minimizes the mean squared error (MSE) of a kernel estimator for nonparametric regression. First, a consistent estimate of the exact MSE is constructed. Then the bandwidth that minimizes the estimate of the MSE is calculated. Sufficient conditions under which this bandwidth is asymptotically optimal and normally distributed are given. The local bandwidth selection procedure was implemented on some simulated data and compared to a global bandwidth selection procedure proposed by Rice (1984b). A 68%–91% reduction in the average MSE of a kernel estimator was realized with the local bandwidth selection procedure. Such a scheme was also studied by Müller (1985) and termed a direct pilot estimator approach. Müller derived sufficient conditions similar to those presented here, under which the direct pilot estimator approach provides a consistent estimator of the local bandwidth. His result is slightly more general in the specification of the interval that is searched for the optimizing bandwidth. The asymptotic normality of the optimizing bandwidth and the simulation results presented here have not to our knowledge appeared in the statistical literature. Müller and Stadtmüller (1987) proposed a local bandwidth selection procedure for kernel estimates that is based on the asymptotic expression for the MSE of the kernel estimator. Our procedure is based upon the finite sample expression for the MSE. It remains to be shown how the two procedures compare when applied to small data sets.