A comparison of the Reinsch and Speckman splines

Abstract

In the context of nonparametric regression, spline smoothers can be used both for the estimation of the curve itself and for the estimation of the error variance. We compare two approaches that have been suggested: the Reinsch spline, i.e. roughness penalty, and the Speckman spline, minimax. The comparison is made on the basis of the summed mean squared error for curve estimation and on the basis of a closely related criterion for variance estimation. We show in both asymptotic and small-sample studies that the Reinsch smoothing spline is close to optimal, both as a curve and as a variance estimator.