A generalized C_p criterion for derivative estimation

The performance of a nonparametric regression method depends on the values chosen for one or more tuning parameters. Although much attention has been given to tuning parameter selection for recovering a mean response function, comparatively few studies have addressed tuning parameter selection for derivative estimation, and most of these studies have focused on a specific nonparametric regression method such as kernel smoothing. In this article, we propose using a generalized C_p (GC_p) criterion to select tuning parameters for derivative estimation. This approach can be used with any nonparametric regression method that estimates derivatives linearly in the observed responses, including but not limited to kernel smoothing, local regression, and smoothing splines. The GC_p criterion is a proxy for the unobservable sum of squared errors in estimating the derivative, and, thus, one can better estimate the derivative by selecting tuning parameters at which GC_p is small. We provide both empirical support for GC_p through simulation studies and theoretical justification in the form of an asymptotic efficiency result. We also describe a motivating practical application in analytic chemistry and assess the capabilities of GC_p in that context. Supplementary materials for this article are available with the on-line journal.