Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Non linear estimation over weak Besov spaces and minimax Bayes method

Auteur(s):

Code(s) de Classification MSC:

• 62C10 Bayesian problems; characterization of Bayes procedures
• 62G05 Estimation
• 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
• 62G20 Asymptotic properties

Résumé: Weak Besov spaces naturally appear in statistics or in approximation theory to measure the performance of classical procedures like wavelet thresholding. The first goal of this paper is to evaluate the minimax risk over the weak Besov balls $\wbe$ and for the ${\cal B}_{s'p'p'}-$loss under the white noise model by using Bayes methods. Under suitable conditions, we show that the rate of convergence for $\wbe$ is the same as for the strong Besov ball ${\cal B}_{s,p,q}(C)$, included into $\wbe$. We exploit the least favorable priors the Bayes method exhibits to build some realizations of the worst functions to be estimated and lying in $\wbe$. Furthermore, we note that these functions cannot belong to ${\cal B}_{s,p,q}(C)$, which provides another motivation for statisticians to consider weak Besov spaces. The second goal of this paper is to explore the Bayes approach to thresholding. For this purpose, we build level-dependent thresholding rules that attain the exact rates of convergence, and whose thresholds are related to the parameters of the least favorable priors.

Mots Clés: Bayes method ; least favorable priors ; minimax risk ; rate of convergence ; thresholding rules ; weak Besov spaces

Date: 2001-03-08

Prépublication numéro: PMA-641

Postscript file : PMA-641.ps