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 mean and variance functions $b$ and $\sigma^2$
in a model:
$X_{i+1}=b(X_i)+\sigma(X_i)\varepsilon_{i+1}$. For this purpose,
we consider a collection of finite dimensional linear spaces . We
estimate $b$ using a mean squares estimator built on a data driven
selected linear space among the collection. Then an analogous
procedure estimates $\sigma^2$, using a possibly different
collection of models. Both data driven choices are performed via
the minimization of penalized mean squares contrasts. The penalty
functions are random in order not to depend on unknown
variance-type quantities. In all cases, we state non asymptotic
risk bounds in $\LL_2$ empirical norm for our estimators and we
show that they are both adaptive in the minimax sense over a large
class of Besov balls. Lastly, we give the results of intensive
simulation experiments which show the good performances of our
estimator.

**Mots Clés:** *Nonparametric regression ; Least-squares estimator ; Adaptive estimation ; Autoregression ; Variance estimation ; Mixing processes*

**Date:** 2001-05-03

**Prépublication numéro:** *PMA-652*

**Pdf file (with figures) : **PMA-652.pdf