On confidence intervals for GAMs based on penalized regression splines

Generalized additive models represented using 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 Silverman’s (1985) re-derivation of the intervals proposed by Wahba (1983) for 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 paper also suggests a simple and efficient simulation scheme for dealing with the poor interval coverage that can sometimes result from conditioning on smoothing parameter estimates.