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We consider the estimation of quadratic functionals in a Gaussian sequencemodel where the eigenvalues are supposed to be unknown and accessible throughnoisy observations only. Imposing smoothness assumptions both on the signal andthe sequence of eigenvalues, we develop a minimax theory for this problem. Wepropose a truncated series estimator and show that it attains the optimal rateof convergence if the truncation parameter is chosen appropriately.Consequences for testing problems in inverse problems are equally discussed: inparticular, the minimax rates of testing for signal detection andgoodness-of-fit testing are derived.

Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems