Minimax Signal Detection in Ill-Posed Inverse Problems

Details Minimax Signal Detection in Ill-Posed Inverse Problems Minimax Signal Detection in Ill-Posed Inverse Problems

2010-01-12

arxiv.org/abs/1001.1853v3

Mathematics

Statistics Theory

56 pages. New version. The results in the previous version have been extended to the case of $l^q$-ellipsoids, $q \in (0,2]$.

Scientific paper

Ill-posed inverse problems arise in various scientific fields such as partial differential equations. We consider the signal detection problem for mildly, severely and extremely ill-posed inverse problems with Sobolev, analytic and generalized analytic classes of functions under the Gaussian white noise model. We study both rate and sharp asymptotics for the error probabilities in the minimax setup. By construction, the derived tests are non-adaptive. Minimax rate-optimal adaptive tests of rather simple structure are also constructed. The minimax signal detection problem for mildly ill-posed inverse problems with Besov classes (bodies) of functions is also considered.

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