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

### Mixed Gaussian white noise

Auteur(s):

Code(s) de Classification MSC:

• 62C20 Minimax procedures
• 62G07 Curve estimation (nonparametric regression, density estimation, etc.)

Résumé: We study the problem of estimating a signal $f$ from noisy data under squared-error loss. We assume that $f$ belongs to a certain Sobolev class. The noise process is represented by $t \rightarrow \frac{1}{\sqrt{n}}\int_0^t \sqrt{V_s}dW_s$, where $V$ is a random process independent of the driving Brownian motion $W$. Thus, conditional on $V$, the function $f$ is observed with Gaussian white noise. This setup generalizes the traditional `ideal signal $+$ noise' framework adopted in nonparametric estimation. We establish upper and lower bounds for the asymptotic minimax risk (as $n \rightarrow \infty$) up to constants. We show in particular that the bound of the Pinsker estimator, which is optimal in the case of a deterministic $V$, can be strictly improved if the law of $V$ is known and non degenerate. We characterize the influence of the law of $V$ on the optimal constants and construct asymptotically efficient estimators. We present some statistical models which lie in the scope of this new estimation procedure.

Mots Clés: Gaussian white noise ; mixed normality ; nonparametric $L_2$ efficiency ; Pinsker bound ; linear filtering ; minimax estimation ; Sobolev ellipsoids

Date: 1999-05-19

Prépublication numéro: PMA-504