Kernels for Nonparametric Curve Estimation

Abstract

SUMMARY The choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated. Explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error, IMSE (the present proof of the optimality of the latter kernels is restricted up to order k = 5). These kernels are also of interest for the nonparametric estimation of probability densities and spectral densities. A finite sample study indicates that higher order kernels – asymptotically improving the rate of convergence – may become attractive for realistic finite sample size. Suitably modified kernels are considered for estimating at the extremities of the data, in a way which allows to retain the order of the bias found for interior points.