Nonparametric estimation when data on derivatives are available

TL;DR

This paper explores how available derivative data can be used to reduce the effective dimensionality in nonparametric estimation, improving convergence rates and estimator efficiency.

Contribution

It introduces a method to replace local averages with nonlocal averages using derivative data, achieving dimension reduction and optimal convergence rates.

Findings

Dimension reduction improves estimation efficiency.

Kernel estimators are effective under the proposed method.

Application to electricity cost estimation demonstrates practical benefits.

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.