Open Access
February 2007 Nonparametric estimation when data on derivatives are available
Peter Hall, Adonis Yatchew
Ann. Statist. 35(1): 300-323 (February 2007). DOI: 10.1214/009053606000001127


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.


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Peter Hall. Adonis Yatchew. "Nonparametric estimation when data on derivatives are available." Ann. Statist. 35 (1) 300 - 323, February 2007.


Published: February 2007
First available in Project Euclid: 6 June 2007

zbMATH: 1114.62049
MathSciNet: MR2332277
Digital Object Identifier: 10.1214/009053606000001127

Primary: 62G08 , 62G20
Secondary: 62P20 , 91B38

Keywords: cost function estimation , Dimension reduction , kernel methods , Nonparametric regression , partial derivative data , rates of convergence , statistical smoothing

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 1 • February 2007
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