Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. e-mail: email@example.com
ON CONFIDENCE INTERVALS FOR GENERALIZED ADDITIVE MODELS BASED ON PENALIZED REGRESSION SPLINES
Article first published online: 19 DEC 2006
Australian & New Zealand Journal of Statistics
Volume 48, Issue 4, pages 445–464, December 2006
How to Cite
Wood, . S. N. (2006), ON CONFIDENCE INTERVALS FOR GENERALIZED ADDITIVE MODELS BASED ON PENALIZED REGRESSION SPLINES. Australian & New Zealand Journal of Statistics, 48: 445–464. doi: 10.1111/j.1467-842X.2006.00450.x
- Issue published online: 19 DEC 2006
- Article first published online: 19 DEC 2006
- Received March 2004; accepted November 2005.
- Bayesian confidence interval;
- generalized additive model (GAM);
- generalized cross validation (GCV);
- multiple smoothing parameters;
- penalized regression spline
Generalized additive models represented using low rank penalized regression splines, estimated by penalized likelihood maximisation and with smoothness selected by generalized cross validation or similar criteria, provide a computationally efficient general framework for practical smooth modelling. Various authors have proposed approximate Bayesian interval estimates for such models, based on extensions of the work of Wahba, G. (1983)[Bayesian confidence intervals for the cross validated smoothing spline. J. R. Statist. Soc. B45, 133–150] and Silverman, B.W. (1985)[Some aspects of the spline smoothing approach to nonparametric regression curve fitting. J. R. Statist. Soc. B47, 1–52] on smoothing spline models of Gaussian data, but testing of such intervals has been rather limited and there is little supporting theory for the approximations used in the generalized case. This paper aims to improve this situation by providing simulation tests and obtaining asymptotic results supporting the approximations employed for the generalized case. The simulation results suggest that while across-the-model performance is good, component-wise coverage probabilities are not as reliable. Since this is likely to result from the neglect of smoothing parameter variability, a simple and efficient simulation method is proposed to account for smoothing parameter uncertainty: this is demonstrated to substantially improve the performance of component-wise intervals.