# Estimation adaptative de fonctionnelles linéaires dans le modèle de convolution

vendredi 12 octobre 2007, 9h30 - 10h45

Travail en collaboration avec{{ Cristina BUTUCEA}}, Université de Lille (UTSL)

{{Abstract}}.
We consider the model $Z_i=X_i+\varepsilon_i$ for i.i.d. $X_i$’s and $\varepsilon_i$’s and independent sequences $(X_i)$, $i\in N$ and $(\varepsilon_i)$, $i \in N$. The density of $\varepsilon$ is assumed to be known whereas the one of $X_1$ denoted by $g$ is unknown. Our aim is to study the estimation of linear functionals of $g$, $<\psi,g>$ for a known function $\psi$. We propose a general estimator of $< \psi,g>$ and study the rate of convergence of its quadratic risk in function of the smoothness of $g$, $f_\varepsilon$ and $\psi$. Different dependency contexts are also considered, allowing to deal with stochastic volatility or ARCH models. An adaptive estimator is then proposed, following a method studied by Laurent {et al.}~(2006) in another context. The quadratic risk of this estimator is studied. The results are applied to adaptive pointwise deconvolution, in which context losses in the adaptive rates are shown to be optimal in the minimax sense. They are also applied in the context of the stochastic volatility model