Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 62G05 Estimation
- 62G20 Asymptotic properties

**Résumé:** 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 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.

**Mots Clés:** *Statistical inverse problems ; Oracle inequalities ; Adaptive curve estimation ; Model selection ; Exact minimax constants*

**Date:** 2000-06-26

**Prépublication numéro:** *PMA-602*

**Postscript file : **PMA-602.ps