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:**

- 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*