Smoothing splines of a fixed order are commonly used as nonparametric regression estimates. The only parameter, then, that needs to be estimated is the smoothing parameter, which is often estimated using some form of cross validation. This work allows the order of the smoothing spline to be estimated using a model in which the order parameter is continuous. Within this setting, generalized cross validation and modified maximum likelihood estimates of the order and smoothing parameters are compared. I show that there are both stochastic and fixed regression functions for which modified maximum likelihood yields asymptotically better estimates of the regression function than generalized cross validation. These results are supported by a small simulation study, although there are functions for which the asymptotic results can be misleading even for fairly large sample sizes.
"Spline Smoothing with an Estimated Order Parameter." Ann. Statist. 21 (3) 1522 - 1544, September, 1993. https://doi.org/10.1214/aos/1176349270