Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems

TL;DR

This paper develops a minimax theory for estimating quadratic functionals in inverse problems with unknown eigenvalues, proposing an optimal truncated estimator and analyzing its implications for signal detection and goodness-of-fit testing.

Contribution

It introduces a minimax framework for quadratic functional estimation with unknown eigenvalues and proposes an optimal truncated estimator under smoothness assumptions.

Findings

The truncated series estimator attains the optimal convergence rate.

Minimax rates for signal detection are derived.

The approach applies to inverse problems with partially unknown operators.

Abstract

We consider the estimation of quadratic functionals in a Gaussian sequence model where the eigenvalues are supposed to be unknown and accessible through noisy observations only. Imposing smoothness assumptions both on the signal and the sequence of eigenvalues, we develop a minimax theory for this problem. We propose a truncated series estimator and show that it attains the optimal rate of convergence if the truncation parameter is chosen appropriately. Consequences for testing problems in inverse problems are equally discussed: in particular, the minimax rates of testing for signal detection and goodness-of-fit testing are derived.