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Optimal regularized hypothesis testing in statistical inverse problems

Remo KretschmannDaniel WachsmuthFrank Werner
Dec 2022
摘要
Testing of hypotheses is a well studied topic in mathematical statistics.Recently, this issue has also been addressed in the context of InverseProblems, where the quantity of interest is not directly accessible but onlyafter the inversion of a (potentially) ill-posed operator. In this study, wepropose a regularized approach to hypothesis testing in Inverse Problems in thesense that the underlying estimators (or test statistics) are allowed to bebiased. Under mild source-condition type assumptions we derive a family oftests with prescribed level $\alpha$ and subsequently analyze how to choose thetest with maximal power out of this family. As one major result we prove thatregularized testing is always at least as good as (classical) unregularizedtesting. Furthermore, using tools from convex optimization, we provide anadaptive test by maximizing the power functional, which then outperformsprevious unregularized tests in numerical simulations by several orders ofmagnitude.
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