Information bounds for inverse problems with application to deconvolution and L\'evy models

TL;DR

This paper develops a framework for understanding the fundamental limits of estimating functionals in inverse problems, especially nonlinear models like deconvolution with Lévy processes, using semiparametric efficiency and information bounds.

Contribution

It introduces a semiparametric efficiency approach for inverse problems, providing a convolution theorem and detailed bounds for nonlinear models like Lévy-based deconvolution.

Findings

Established a general information bound for inverse problems.

Proved a convolution theorem applicable to nonlinear models.

Derived specific bounds for Lévy process deconvolution.

Abstract

If a functional in an inverse problem can be estimated with parametric rate, then the minimax rate gives no information about the ill-posedness of the problem. To have a more precise lower bound, we study semiparametric efficiency in the sense of H\'ajek-Le Cam for functional estimation in regular indirect models. These are characterized as models that can be locally approximated by a linear white noise model that is described by the generalized score operator. A convolution theorem for regular indirect models is proved. This applies to a large class of statistical inverse problems, which is illustrated for the prototypical white noise and deconvolution model. It is especially useful for nonlinear models. We discuss in detail a nonlinear model of deconvolution type where a L\'evy process is observed at low frequency, concluding an information bound for the estimation of linear functionals of the jump measure.