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 λ) perform poorly. Adaptive smoothing splines adopt a variable smoothing parameter λ(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 λ(x)∈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.

Keywords:

• adaptive smoothing splines
• kernel smoothing
• optimal bandwidth
• optimal penalty function

AMS Subject Classification :

• 62G08
• 62G05

Acknowledgements

The authors thank the two referees and the associate editor whose helpful comments greatly helped improving the clarity and presentation of this paper. This work was partially supported by NSF grants 0604736, 0700152, and 0831300