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

### Adaptative estimation of the spectrum of a stationary Gaussian sequence

Auteur(s):

Code(s) de Classification MSC:

• 62G07 Curve estimation (nonparametric regression, density estimation, etc.)
• 62J02 General nonlinear regression

Résumé: In this paper, we study the problem of non parametric estimation of the spectral density $f$ of a stationary Gaussian sequence. For this purpose, we consider a collection of finite dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on possibly irregular grids or spaces of trigonometric polynomials). We estimate the spectral density by a projection estimator based on the periodogram and built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized projection contrast. The penalty function depends on $\|f\|_{\infty}$ but we give results including the estimation of this bound. Moreover, we give extensions to the case of unbounded spectral densities (long memory processes). In all cases, we state non asymptotic risk bounds in I$\!$L$_2$-norm for our estimator and we show that it is adaptive in the minimax sense over a large class of Besov balls.

Mots Clés: Adaptive estimation ; projection estimator ; penalty function ; stationary sequence ; long memory process

Date: 1999-09-15

Prépublication numéro: PMA-525