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:**

- 41A40 Saturation
- 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
- 62G20 Asymptotic properties

**Résumé:** The goal of this paper is to address the following question
: given an estimation method and a prescribed rate of convergence for a given loss function what is the maximal space over which this rate is attained.
We will discuss here the existence and the nature of
maximal spaces in the context of non linear methods based on thresholding procedures.
It turns out that the maximal spaces in this setting will coincide with known smoothness classes. These classes also appear to play an essential role
in approximation theory. We mainly investigate losses which can be expressed as weighted $l_p$-norms of wavelet coefficents (such as particular Besov norms),
or the more difficult case of $L_p$-norms.

**Mots Clés:** *Maximal space ; shrinkage method ; non linear approximation*

**Date:** 1999-10-28

**Prépublication numéro:** *PMA-537*