The Blessing of Derivatives in Nonparametric Estimation

• Activity Number: 260
• Type: Contributed
• Date/Time: Monday, August 1, 2016 : 2:00 PM to 3:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Xiaowu Dai* and Grace Wahba and Peter Qian
• Companies: University of Wisconsin - Madison and University of Wisconsin - Madison and University of Wisconsin - Madison
• Keywords: Nonparametric regression ; partial derivative data ; dimension reduction ; reproducing kernel Hilbert space ; rates of convergence ; radial basis functions

Abstract

We study smoothing regularization methods for incorporating derivatives in nonparametric function estimation. Data with derivative info arise in economics, engineering, uncertainty quantification and many other fields. We obtained new results to show that the dimension of the estimation of a multidimensional function can be reduced if the first-order partial derivatives of the function are available. Also, we established that the regularization with incorporation of derivative data is rate-optimal and adapts to unknown smoothness up to an order related to the given kernel. We provide theoretical results to show that in finite sample cases, the proposed regularization estimator produces smaller mean squared error than least squares estimators used in previous studies. The proposed estimation procedure is easy to implement and generally applicable to a wide range of kernels. Numerical examples are provided to corroborate the derived theoretical results.