On adaptive inverse estimation of linear functionals in Hilbert scales

Abstract

We address the problem of estimating the value of a linear functional $local asymptotic normalitygle f,x \rangle$ from random noisy observations of $y=Ax$ in Hilbert scales. Both the white noise and density observation models are considered. We propose an estimation procedure that adapts to unknown smoothness of $x$, of $f$, and of the noise covariance operator. It is shown that accuracy of this adaptive estimator is worse only by a logarithmic factor than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered.