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 $(X_i)_{i∈ℕ}$ and $(\varepsilon_i)_{i∈ℕ}$. The density $f_\varepsilon$ of $\varepsilon_1$ is assumed to be known, whereas the one of $X_1$, denoted by $g$, is unknown. Our aim is to estimate linear functionals of $g, 〈ψ, g〉$ for a known function $ψ$. We propose a general estimator of $〈ψ, g〉$ and study the rate of convergence of its quadratic risk as a function of the smoothness of $g, f_\varepsilon$ and $ψ$. Different contexts with dependent data, such as stochastic volatility and AutoRegressive Conditionally Heteroskedastic models, are also considered. An estimator which is adaptive to the smoothness of unknown $g$ is then proposed, following a method studied by Laurent et al. (Preprint (2006)) in the Gaussian white noise model. We give upper bounds and asymptotic lower bounds of the quadratic risk of this estimator. 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.