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:**

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

**Résumé:** In this paper, we provide a new way of evaluating the performances of a
statistical estimation
procedure. This point of view consists in
investigating the maximal set where a procedure has a given
rate of convergence. Although the setting is not extremely
different from the minimax context, it is less pessimistic
and provides a functional set which is authentically connected
to the procedure and the model.
We also investigates more traditional concerns about
procedures: oracle inequalities.
This notion becomes more
difficult even to be practically defined when the loss function
is not the $\bL_2$-norm.
We explain the difficulties arising there,
and suggest a new definition, in
the cases of $\bL_p$-norms and point-wise estimation.
The connections between maxisets and local oracle inequalities are
investigated: we prove that
verifying a local oracle inequality implies that the maxiset
automatically contains a prescribed set linked with the oracle inequality.
We have investigated the consequences of the previous statement on
well known
efficient adaptive methods: Wavelet thresholding and local bandwidth
selection.
We can prove local oracle inequalities for these methods
and draw the conclusions about there associated maxisets.

**Mots Clés:** *Non parametric estimation ; denoising ; minimax rate of convergence ;
oracle inequalities ; saturation spaces ; wavelet thresholding ; local bandwidth selection*

**Date:** 2000-01-05

**Prépublication numéro:** *PMA-556*