Locally optimal adaptive smoothing splines

Abstract

Smoothing splines are widely used for estimating an unknown function in the nonparametric regression. If data have large spatial variations, however, the standard smoothing splines (which adopt a global smoothing parameter lambda) perform poorly. Adaptive smoothing splines adopt a variable smoothing parameter lambda(x) (i.e. the smoothing parameter is a function of the design variable x) to adapt to varying roughness. In this paper, we derive an asymptotically optimal local penalty function for lambda(x) is an element of C-3 under suitable conditions. The derived locally optimal penalty function in turn is used for the development of a locally optimal adaptive smoothing spline estimator. In the numerical study, we show that our estimator performs very well using several simulated and real data sets.