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:** We consider the problem of estimating an unknown function
$f$ in a regression setting with random design. Instead of expanding the
function on a regular wavelet basis, we expand it on the basis $\{\psi_{jk}(G),
j,\; k\}$ warped with the design. This allows to perform a very stable and
computable thresholding algorithm. We investigate the properties of this new
basis. In particular, we prove that if the design has a property of
Muckenhoupt type, this new basis has a behavior quite similar to a regular
wavelet basis. This enables us to prove that the associated thresholding
procedure achieves rates of convergence which have been proved to be
minimax in the uniform design case.

**Mots Clés:** *nonparametric regression ; random design ; wavelet thresholding ; warped wavelets ; maxisets ; Muckenhoupt weights*

**Date:** 2003-01-24

**Prépublication numéro:** *PMA-788*

**Pdf file: **PMA_788.pdf