Asymptotically Optimal Estimators of General Regression Functionals

Let X, Y be random vectors with values in Rd and R1, respectively, and denote by F(· | x) the conditional distribution function of Y given X = x. It is well known that the kernel estimator of the regression functional ϑ(x) ≔ T(F(· | x)), based on n independent replicates of (X, Y), has optimal asymptotic accuracy in case of T being the mean value and quantile functional. In this paper we discuss conditions under which the kernel estimator has optimal asymptotic accuracy, locally and globally, for a general class of functionals T, containing mean and quantile as particular examples. A weak convergence result for the maximum error over a compact interval completes the paper.