We propose a simple framework for estimating derivatives without cutting the regression function in nonparametric regression. Unlike most existing methods that use the symmetric difference quotients, our method is constructed as a linear combination of observations. It is hence very flexible and applicable to both interior and boundary points, including most existing methods as special cases of ours. Within this framework, we define the variance-minimizing estimators for any order derivative of the regression function with a fixed bias-reduction level. For the equidistant design, we derive the asymptotic variance and bias of these estimators. We also show that our new method will, for the first time, achieve the asymptotically optimal convergence rate for difference-based estimators. Finally, we provide an effective criterion for selection of tuning parameters and demonstrate the usefulness of the proposed method through extensive simulation studies of the firstand second-order derivative estimators.
|Original language||English (US)|
|Journal||Journal of Machine Learning Research|
|State||Published - Aug 1 2016|
Bibliographical noteKAUST Repository Item: Exported on 2021-07-07
Acknowledgements: The authors thank the editor, the associate editor and the two referees for their constructive comments that led to a substantial improvement of the paper. The work of Wenlin Dai and Marc G. Genton was supported by King Abdullah University of Science and Technology (KAUST). Tiejun Tong’s research was supported in part by Hong Kong Baptist University FRG grants FRG1/14-15/044, FRG2/15-16/038, FRG2/15-16/019 and FRG2/14-15/084.
- Linear combination
- Nonparametric derivative estimation
- Nonparametric regression
- Optimal sequence
- Taylor expansion
ASJC Scopus subject areas
- Control and Systems Engineering
- Statistics and Probability
- Artificial Intelligence