A framework to select tuning parameters for nonparametric derivative estimation

In this paper, we propose a general framework to select tuning parameters for the nonparametric derivative estimation. The new framework broadens the scope of the previously proposed generalized $C_p$ criterion by replacing the empirical derivative with any other linear nonparametric smoother. We provide the theoretical support of the proposed derivative estimation in a random design and justify it through simulation studies. The practical application of the proposed framework is demonstrated in the study of the age effect on hippocampal gray matter volume in healthy adults from the IXI dataset and the study of the effect of age and body mass index on blood pressure from the Pima Indians dataset.