Asymptotically optimal selection of a piecewise polynomial estimator of a regression function

Let (X, Y) be a pair of random variables such that X = (X1,..., Xd) ranges over a nondegenerate compact d-dimensional interval C and Y is real-valued. Let the conditional distribution of Y given X have mean [theta](X) and satisfy an appropriate moment condition. It is assumed that the distribution of X is absolutely continuous and its density is bounded away from zero and infinity on C. Without loss of generality let C be the unit cube. Consider an estimator of [theta] having the form of a piecewise polynomial of degree kn based on mnd cubes of length 1/mn, where the mnd(dkn+d) coefficients are chosen by the method of least squares based on a random sample of size n from the distribution of (X, Y). Let (kn, mn) be chosen by the FPE procedure. It is shown that the indicated estimator has an asymptotically minimal squared error of prediction if [theta] is not of the form of piecewise polynomial.