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In the inverse Gaussian sequence space model with additional noisyobservations of the operator, we derive nonasymptotic minimax radii of testingfor ellipsoid-type alternatives simultaneously for both the signal detectionproblem (testing against zero) and the goodness-of-fit testing problem (testingagainst a prescribed sequence) without any regularity assumption on the nullhypothesis. The radii are the maximum of two terms, each of which only dependson one of the noise levels. Interestingly, the term involving the noise levelof the operator explicitly depends on the null hypothesis and vanishes in thesignal detection case. The minimax radii are established by first showing alower bound for arbitrary null hypotheses and noise levels. For the upper boundwe consider two testing procedures, a direct test based on estimating theenergy in the image space and an indirect test. Under mild assumptions, weprove that the testing radius of the indirect test achieves the lower bound,which shows the minimax optimality of the radius and the test. We highlight theassumptions under which the direct test also performs optimally. Furthermore,we apply a classical Bonferroni method for making both the indirect and thedirect test adaptive with respect to the regularity of the alternative. Theradii of the adaptive tests are deteriorated by an additional log-factor, whichwe show to be unavoidable. The results are illustrated considering Sobolevspaces and mildly or severely ill-posed inverse problems.

Adaptive minimax testing in inverse Gaussian sequence space models