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

### Regression in random design and warped wavelets

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