Hypothesis testing under composite functions alternative

In this paper, we consider the problem of the minimax hypothesis testing in the multivariate white gaussian noise model. We want to test the hypothesis about the absence of the signal against the alternative belonging to the set of smooth composite functions separated away from zero in sup-norm. We propose the test procedure and show that it is optimal in view of the minimax criterion if the smoothness parameters of the composition obey some special assumption. In this case we also present the explicit formula for minimax rate of testing. If this assumption does not hold, we give the explicit upper and lower bounds for minimax rate of testing which differ each other only by some logarithmic factor. In particular, it implies that the proposed test procedure is "almost" minimax. In both cases the minimax rate of testing as well as its upper and lower bounds are completely determined by the smoothness parameters of the composition.