Choosing a smoothing parameter by minimizing an estimate of the risk for shape constrained smoothing splines

The problem of choosing a smoothing parameter for shape constrained smoothing splines is considered. It is shown that the fitted curve can be written as a positive linear combination of edges of the polyhedral constraint cone, and that this form provides an estimate of the risk, or expected squared error loss. The smoothing parameter is chosen by minimizing this estimate over a range of parameters.