Nonparametric estimation when data on derivatives are available

The Annals of Statistics · 2007 · Volume 35 · Issue 1

Abstract

We consider settings where data are available on a nonparametric function and various partial derivatives. Such circumstances arise in practice, for example in the joint estimation of cost and input functions in economics. We show that when derivative data are available, local averages can be replaced in certain dimensions by nonlocal averages, thus reducing the nonparametric dimension of the problem. We derive optimal rates of convergence and conditions under which dimension reduction is achieved. Kernel estimators and their properties are analyzed, although other estimators, such as local polynomial, spline and nonparametric least squares, may also be used. Simulations and an application to the estimation of electricity distribution costs are included.

Keywords

62G08, 62G20, 62P20, 91B38, cost function estimation, Dimension reduction, kernel methods, Nonparametric regression, partial derivative data, rates of convergence, statistical smoothing

Enable JavaScript for citation tools and additional paper interactions.