Adaptive estimation of linear functionals in the convolution model and applications.

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 $(X_i)_{i\in {\mathbb N}}$ and $(\varepsilon_i)_{i\in {\mathbb N}}$. 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 $g$, $\langle \psi,g\rangle$ for a known function $\psi$. We propose a general estimator of $\langle \psi,g\rangle$ 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. An adaptive estimator is then proposed, following a method studied by Laurent {\it 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 to pointwise Laplace transform estimation in the standard context and in the context of the stochastic volatility model. Estimation in the context of ARCH-type models lastly illustrates the method.
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Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2009, 15 (1), pp.69-98. 〈10.3150/08-BEJ146〉
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Cristina Butucea, Fabienne Comte. Adaptive estimation of linear functionals in the convolution model and applications.. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2009, 15 (1), pp.69-98. 〈10.3150/08-BEJ146〉. 〈hal-00133750〉

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