Kernel estimation of regression functions

Abstract

For the nonparametric estimation of regression functions with a one-dimensional design parameter, a new kernel estimate is defined and shown to be superior to the one introduced by Priestley and Chao (1972). The results are not restricted to positive kernels, but extend to classes of kernels satisfying certain moment conditions. An asymptotically valid solution for the boundary problem, arising for non-circular models, is found, and this allows the derivation of the asymptotic integrated mean square error. As a special case we obtain the same rates of convergence as for splines. For two optimality criteria (minimum variance, minimum mean square error) higher order kernels are explicitly tabulated.

Research undertaken within project B 1 of the Sonderforschungsbereich 123 (Stochastic Mathematical Models) financed by the Deutsche Forschungsgemeinschaft.

Preliminary results have been obtained in the second author’s diploma at the University of Heidelberg, autumn 1978.