A critical review of univariate non-parametric estimation of first derivatives

This paper gives new guidance for selection of univariate non-parametric derivative estimators with non-iid errors in finite samples. It is shown via an extensive set of Monte Carlo simulations that the generalized $C_P$ criterion of Charnigo et al. (A generalized $C_P$ criterion for derivative estimation. Technometrics. 2011;53:238–253.) with spline smoothing performs the best in situations with minimal noise. For increased noise, generalized cross-validation of Craven and Wahba (Smoothing noisy data with spline functions. Numer Math. 1978;31:377–403) with P spline smoothing and the improved Akaike Information Criterion of Hurvich et al. (Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion. J R Stat Soc Ser B. 1998;60:271–293.) with P spline smoothing are preferred. In the class of kernel smoothing and local regression methods, the local-cubic estimator of Henderson et al. (Gradient-based smoothing parameter selection for nonparametric regression estimation. J Econom. 2015;184:233–241.) generally outperforms its competitors. An internal meta-analysis separately favours the generalized $C_P$ method and P spline smoothing. The empirical example given provides support for use of the local-cubic estimator.