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:**

- 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