Estimation and detection of functions from anisotropic Sobolev classes

Abstract

We consider the problems of estimating and detecting an unknown function f depending on a multidimensional variable (for instance, an image) observed in the Gaussian white noise. It is assumed that f belongs to anisotropic Sobolev class. The case of a function of infinitely many variables is also considered. An asymptotic study (as the noise level tends to zero) of the estimation and detection problems is done. In connection with the estimation problem, we construct asymptotically minimax estimators and establish sharp asymptotics for the minimax integrated squared risk. In the detection problem, we construct asymptotically minimax tests and provide conditions for distinguishability in the problem.