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 estimating and
detecting unknown signal, the problems of rate and sharp optimality (minimaxity)
are investigated.
In the case when $d$ is fixed and finite
the sharp optimality problems are solved.
In the infinite-dimensional case log-asymptotics of the risk in the estimation problem
and of the separation rate in the detection problem are obtained.
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