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Biometrika 2002 89(2):251-263; doi:10.1093/biomet/89.2.251
© 2002 by Biometrika Trust
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Degrees-of-freedom tests for smoothing splines

Eva Cantoni1 and Trevor Hastie2

1 Department of Econometrics, University of Geneva, 40, Boulevard du Pont d'Arve, 1211 Geneva 4, Switzerland eva.cantoni@metri.unige.ch 2 Statistics Department, Stanford University, Sequoia Hall–390 Serra Mall, Stanford, California 94305, U.S.A.hastie@stat.stanford.edu

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.

Key Words: Additive model; Degrees-of-freedom test; Smoothing parameter selection; Smoothing spline

Received April 2001. Revised August 2001


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