Local Polynomial Fitting: A Standard for Nonparametric Regression!

Among the various nonparametric regression methods, weighted local polynomial fitting is the one which is gaining increasing popularity. This is due to the attractive minimax efficiency of the method and to some further desirable properties such as the automatic incorporation of boundary treatment. In this paper previous results are extended in two directions: in the one-dimensional case, not only local linear fitting is considered but also polynomials of other orders and estimating derivatives. In addition to deriving minimax properties, optimal weighting schemes are derived and the solution obtained at the boundary is discussed in some detail. An equivalent. kernel formulation serves as a tool to derive many of these properties. In the higher dimensional case local linear fitting is considered. Properties in terms of minimax efficiency are derived and optimal weighting ISupported by the Deutsche Forschungsgemeinschaft. 2Supported by NSF grant DMS-9203135 We would like to thank Burkhardt Seifert for intellectual support, in particular in the improvement of section 3.