Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

Abstract

The aim of this paper is to establish non-asymptotic minimax rates for goodness-of-fit hypotheses testing in an heteroscedastic setting. More precisely, we deal with sequences (Y j) j∈j of independent Gaussian random variables, having mean (θ j) j∈j and variance (σ j) j∈j. The set J will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to l 2 norm, without assumption on (σ j) j∈j and on several functions spaces. Our point of view is entirely non-asymptotic.