Some asymptotic results on generalized penalized spline smoothing

Göran Kauermann, Tatyana Krivobokova and Ludwig Fahrmeir

Journal Of The Royal Statistical Society Series B, 2009, vol. 71, issue 2, pages 487-503

Abstract: The paper discusses asymptotic properties of penalized spline smoothing if the spline basis increases with the sample size. The proof is provided in a generalized smoothing model allowing for non-normal responses. The results are extended in two ways. First, assuming the spline coefficients to be "a priori" normally distributed links the smoothing framework to generalized linear mixed models. We consider the asymptotic rates such that the Laplace approximation is justified and the resulting fits in the mixed model correspond to penalized spline estimates. Secondly, we make use of a fully Bayesian viewpoint by imposing an "a priori" distribution on all parameters and coefficients. We argue that with the postulated rates at which the spline basis dimension increases with the sample size the posterior distribution of the spline coefficients is approximately normal. The validity of this result is investigated in finite samples by comparing Markov chain Monte Carlo results with their asymptotic approximation in a simulation study. Copyright (c) 2009 Royal Statistical Society.

Date: 2009
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