Smoothing splines with a (generalized) cross-validated smoothing parameter have been a popular tool in nonparametric statistics since works of Grace Wahba and co-authors. It also has been known that the smoothing spline estimator can be obtained as a best linear predictor under the assumption that the underlying mean function of the data is the realization of a stochastic process. In this case the unknown smoothing parameter can be estimated from the corresponding likelihood. A rigorous comparison of both smoothing parameters (cross-validated and maximum likelihood one) has been done so far only in the stochastic framework, that is under the assumption that the data are in fact a realization of some random process. In this work we extend these results in two ways. First, we consider general low-rank spline estimators (known as penalized splines), where not only the smoothing parameter, but also the number of knots controls the fit. Second, we compare both smoothing parameters also in the fixed framework, that is assuming that the true mean function comes from a certain class of smooth functions. We find that the likelihood based smoothing parameter has in general considerably smaller variance, than the cross-validated one in both frameworks, depending on the number of knots and degree of the spline. However, in the fixed framework the maximum likelihood based estimator is biased with respect to the mean squared error minimizer, again depending on the number of knots, signal-to-noise ratio and degree of the spline used. We illustrate our theoretical findings with a simulation study and real-data example.
Keywords: penalized splines; smoothing parameter; mean squared error minimizer
Biography: Obtained PhD from the Bielefeld University, Germany in 2007. After a one year postdoc at the University of Leuven, Belgium, took the position of assistant professor at the University of Goettingen, Germany. In June 2010 promoted to associate professor with tenure track.