Optimal weighting for linear inverse problems

Abstract

Linear equations in functional spaces where the solution is not continuous require regularization to estimate the unknown function of interest. In this paper we consider the estimation of an infinite dimensional parameter φ by solving a linear equation $\hat{r}=K\mathit{\phi }+U$, where the random noise U has a variance Σ. Under this set-up, we derive the optimal weighting operator which minimizes the mean integrated square error (MISE). In the finite dimensional case the minimum variance estimator is obtained by weighting the equation by ${\mathrm{\Sigma }}^{-1∕2}$. However in the infinite dimensional case that we consider, regularization introduces a bias to the estimator. We show that in the infinite dimensional case, the optimal estimator in terms of the MISE should involve Σ and the unknown smoothness of φ. We then use this result to propose a new feasible two-step estimator. We illustrate our theoretical findings and the small sample properties of the proposed optimal estimator by means of simulations.