MODEL INDEXING AND SMOOTHING PARAMETER SELECTION IN NONPARAMETRIC FUNCTION ESTIMATION

Smoothing parameter selection is among the most intensively studied su bjects in nonparametric function estimation. A closely related issue, that of identifying a proper index for the smoothing parameter, is how ever largely neglected in the existing literature. Through heuristic a rguments and simple simulations, we show that most current working ind ices are conceptually ''incorrect'', in the sense that they are not in terpretable across-replicate in repeated experiments. As a con sequenc e, a few popular working concepts, such as expected mean square error and ''degrees of freedom'', appear vulnerable under close scrutiny. Du e to technical constraints, the arguments are mainly developed in the penalized likelihood setting, but conceptual parallels can be drawn to other settings as well. In the light of our findings, simulations and discussion are also presented to compare the relative merits of the s imple cross-validation method versus the more sophisticated plug-in me thod for smoothing parameter selection, and to explore related issues. The development stems from an attempt to understand the well-publiciz ed negative correlation between optimal and cross-validation smoothing parameters, which however turns out to bear little statistical releva nce.