Nonparametric estimation when data on derivatives are available. (English) Zbl 1114.62049

Summary: 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.


62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62P20 Applications of statistics to economics
91B38 Production theory, theory of the firm


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