Nonparametric estimators which can be "plugged-in"

Abstract

We consider nonparametric estimation of an object such as a probability density or a regression function. Can such an estimator achieve the ratewise minimax rate of convergence on suitable function spaces, while, at the same time, when "plugged-in," estimate efficiently (at a rate of~$n^{-1/2}$ with the best constant) many functionals of the object? For example, can we have a density estimator whose definite integrals are efficient estimators of the cumulative distribution function? We show that this is impossible for very large sets, for example, expectations of all functions bounded by $M<\infty$. However, we also show that it is possible for sets as large as indicators of all quadrants, that is, distribution functions. We give appropriate constructions of such estimates.