The smoothing parameter, confidence interval and robustness for smoothing splines

Abstract

Diagnostic measure for nonparametric regression using splines is given. The measure which incorporates important information provided by the smoothing parameter has the potential of identifying ‘unusual’ observations. These influential observations can substantially influence the global behavior of the fitted curve. In addition, the case influence on the inference regions for the curve is also discussed. A robust nonparametric procedure can be developed by downweighting these influential observations. The optimal properties related to the proposed robust method are proven. Numerical example and simulation results illustrate the techniques. Several applications are also given for a variety of nonparametric regression models.

Keywords:

• Generalized cross-validation
• Influential observations
• Penalized likelihood
• Regression diagnostics
• Robustness
• Smoothing splines

Acknowledgements

The author would like to thank a referee, an associate editor and the journal editor for helpful suggestions that led to a substantial improvement in this paper. Thanks also to Professor Kosorok at Madison, USA, for helpful comments on an earlier version of this manuscript. This research is partly supported by Taiwan NSC Grant (Project: NSC 93-2118-M-029-002).