Degrees-of-freedom tests for smoothing splines

When using smoothing splines to estimate a function, the user faces the problem of choosing the smoothing parameter. Several techniques are available for selecting this parameter according to certain optimality criteria. Here, we take a different point of view and we propose a technique for choosing between two alternatives, for example allowing for two different levels of degrees of freedom. The problem is addressed in the framework of a mixed-effects model, whose assumptions ensure that the resulting estimator is unbiased. A likelihood-ratio-type test statistic is proposed, and its exact distribution is derived. Tests of linearity and overall effect follow directly. We then extend this idea to additive models where it provides a more attractive alternative than multi-parameter optimisation, and where it gives exact distributional results that can be used in an analysis-of-deviance-type approach. Examples on real data and a simulation study of level and power complete the paper.