  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.
