Preaveraged Localized Orthogonal Polynomial Estimators for Surface Smoothing and Partial Differentiation

Abstract

Abstract We propose a multivariate smoothing method based on products of localized orthogonal polynomial series estimators for a smooth regression surface in the fixed-design regression model. The estimation of partial derivatives is included. The proposed method provides for automatic and efficient boundary modifications near the edges of the surface, assuming that the boundary of the support of the regression function satisfies some regularity conditions. By allowing for a preaveraging step, the corresponding algorithms are speeded up considerably and are easy to implement. Computation of special boundary kernels, as required by the kernel method to avoid edge effects, is not necessary. It is shown that under sufficient smoothness assumptions, the global average mean squared error has the same optimal rate of convergence as the mean squared error at an interior point; that is, the boundary correction is asymptotically effective. The method depends on two smoothing parameters, one determining the amount ...