Sharp Adaptive Estimation of Linear Functionals

Abstract

We consider estimation of a linear functional $T(f)$ where $f$ is an unknown function observed in Gaussian white noise.We find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations. The results allow evaluating explicitly the effect of dimension and treating general scales of classes. Furthermore, we establish a connection between sharp adaptation and optimal recovery. Namely, we propose a scheme that reduces the construction of sharp adaptive estimators on a scale of functional classes to a solution of the corresponding optimization problem.