Estimating the accuracy of (local) cross-validation via randomised GCV choices in kernel or smoothing spline regression

In nonparametric regression, it is generally crucial to select ‘nearly’ optimal smoothing parameters for which the underlying average squared error (Δ ), with given weights, is ‘nearly’ minimised. The cross-validation (CV) selector or the GCV selector are popular for this task, but it has been observed by many statisticians that these selectors may happen to be ‘not sufficiently’ accurate in some situations. So a practical matter of great importance is the development of reliable estimates of this accuracy. The purpose of this paper is to show that the simulation of the randomised GCV selector or a simple general variant using an ‘augmented-randomised-trace’, can provide useful inferences, like consistent estimates of the standard error in the CV selector or of the expected increase of Δ due to this error. Furthermore, this also provides a tool for constructing more parsimonious curve estimates having almost the same asymptotic justification as the CV estimate, namely with similar increase of Δ up to a given factor. Rigorous proofs are given in the context of one-dimensional kernel regression. Simulated examples, also in this context, illustrate the usefulness of the methodology even at moderate sample sizes. Some direct extensions (for multi-dimensional kernels, equispaced splines) of the theoretical results are outlined. We give heuristics which indicate that the general methodology proposed in this article should be useful in many curve-, surface- or image-estimation problems when using spline-like smoothers.