Abstract: Smoothing parameter selection is among the most intensively studied subjects in nonparametric function estimation. A closely related issue, that of identifying a proper index for the smoothing parameter, is however largely neglected in the existing literature. Through heuristic arguments and simple simulations, we show that most current working indices are conceptually ``incorrect'', in the sense that they are not interpretable across-replicate in repeated experiments. As a con sequence, a few popular working concepts, such as expected mean square error and ``degrees of freedom'', appear vulnerable under close scrutiny. Due 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 simple cross-validation method versus the more sophisticated plug-in method for smoothing parameter selection, and to explore related issues. The development stems from an attempt to understand the well-publicized negative correlation between optimal and cross-validation smoothing parameters, which however turns out to bear little statistical relevance.
Key words and phrases: Constraint, cross-validation, kernel method, negative correlation, penalized likelihood, plug-in method.