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.)
- 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*