Nonparametric estimators which can be "plugged-in"

Ya'acov Ritov, Peter J. Bickel
2003 Annals of Statistics  
We consider nonparametric estimation of an object such as a probability density or a regression function. Can such an estimator achieve the 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
more » ... ow that this is impossible for very large sets, e.g., expectations of all functions bounded by M < ∞. However we also show that it is possible for sets as large as indicators of all quadrants, i.e., distribution functions. We give appropriate constructions of such estimates.
doi:10.1214/aos/1059655904 fatcat:iewyg4e4unad7divhxxrlo4mwe