Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes

Abstract

The problem of nonparametric function estimation in the Gaussian white noise model is considered. It is assumed that the unknown function belongs to one of the Sobolev classes, with an unknown regularity parameter. Asymptotically exact adaptive estimators of functions are proposed on the scale of Sobolev classes, with respect to pointwise and sup-norm risks. It is shown that, unlike the case of $L_2$-risk, a loss of efficiency under adaptation is inevitable here. Bounds on the value of the loss of efficiency are obtained.