Abstract

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 of linear estimators be given. Our aim is to mimic the estimator in that has the smallest risk on the true f . Under general conditions, we show that this can be achieved by simple minimization of 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 be also used to obtain sharp minimax adaptive results. In particular, we apply it to show that minimax adaptation on ellipsoids in multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal non-adaptive procedures. Mathematics Subject Classifications: 62G05, 62G20 Key Words: Statistical inverse problems, Oracle inequaliti...

Citations