Adaptive Estimation of the Integral of Squared Regression Derivatives

A problem of estimating the integral of a squared regression function and of its squared derivatives has been addressed in a number of papers. For the case of a heteroscedastic model where smoothness of the underlying regression 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, estimation when neither the degree of regression function smoothness nor design density, scale function and distribution of errors are known. After a brief review of main developments in non‐parametric estimation of non‐linear functionals, we suggest a simple adaptive estimator for the integral of a squared regression function and its derivatives and prove that it is sharp‐optimal whenever the estimated derivative is sufficiently smooth.