Nonparametric estimation when data on derivatives are available

Details Nonparametric estimation when data on derivatives are available Nonparametric estimation when data on derivatives are available

2007-08-03

arxiv.org/abs/0708.0506v1

Annals of Statistics 2007, Vol. 35, No. 1, 300-323

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000001127 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

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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