Open Access
June 2002 Oracle inequalities for inverse problems
L. Cavalier, G. K. Golubev, D. Picard, A. B. Tsybakov
Ann. Statist. 30(3): 843-874 (June 2002). DOI: 10.1214/aos/1028674843


We consider a sequence space model of statistical linear inverse problems where we need to estimate a function $f$ from indirect noisy observations. Let a finite set $\Lambda$ of linear estimators be given. Our aim is to mimic the estimator in $\Lambda$ that has the smallest risk on the true $f$. Under general conditions, we show that this can be achieved by simple minimization of an unbiased risk estimator, provided the singular values of the operator of the inverse problem decrease as a power law. The main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact. This inequality can also be used to obtain sharp minimax adaptive results. In particular, we apply it to show that minimax adaptation on ellipsoids in the multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal nonadaptive procedures.


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L. Cavalier. G. K. Golubev. D. Picard. A. B. Tsybakov. "Oracle inequalities for inverse problems." Ann. Statist. 30 (3) 843 - 874, June 2002.


Published: June 2002
First available in Project Euclid: 6 August 2002

zbMATH: 1029.62032
MathSciNet: MR1922543
Digital Object Identifier: 10.1214/aos/1028674843

Primary: 62G05 , 62G20

Keywords: Adaptive curve estimation , exact minimax constants , Model selection , Oracle inequalities , Statistical inverse problems

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.30 • No. 3 • June 2002
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