Asymptotic efficiency of inverse estimators

Inverse estimation concerns the recovery of an unknown input signal from blurred observations on a known transformation of that signal. The estimators considered in this paper are based on a regularized inverse of the transformation involved, employing a Hilbert space set-up. We focus on properties related to weak convergence. It is shown that linear functionals can be efficiently estimated in the Hájek–LeCam sense, provided they remain restricted to a suitable class. Outside this class, rates different from $\sqrt{n}$ are possible. By way of an example we present the ‘`convolution theorem’ for a deconvolution.