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

### Adaptive prediction and estimation in linear regression with infinitely many parameters

Auteur(s):

Code(s) de Classification MSC:

• 62G05 Estimation
• 62G20 Asymptotic properties

Résumé: The problem of adaptive prediction and estimation in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that is sharp asymptotically minimax adaptive over ellipsoids in $\ell_2$. The method consists in an application of blockwise Stein's rule with weakly'' geometrically increasing blocks to the penalized least squares fits of the first $N$ coefficients. To prove the results we develop oracle inequalities for sequence model with correlated data.

Mots Clés: Linear regression with infinitely many parameters ; adaptive prediction ; exact asymptotics of minimax risk ; blockwise Stein's rule ; oracle inequalities

Date: 2000-06-07

Prépublication numéro: PMA-598

Postscript file : PMA-598.ps