Adaptive estimation of the integral of squared regression derivatives

A problem of estimating the integral of a squared regression function and o f its squared derivatives has been addressed in a number of papers. For the case of a heteroscedastic model where smoothness of the underlying regress ion function, the design density, and the variance of errors are known, the asymptotically sharp minimax lower bound and a sharp estimator were found in Pastuchova & Khasminski (1989). However, there are apparently no results on the either rate optimal or sharp optimal adaptive, or data-driven, esti mation when neither the degree of regression function smoothness nor design density, scale function and distribution of errors are known. After a brie f review of main developments in non-parametric estimation of non-linear fu nctionals, we suggest a simple adaptive estimator for the integral of a squ ared regression function and its derivatives and prove that it is sharp-opt imal whenever the estimated derivative is sufficiently smooth.