Asymptotic minimax rates for abstract linear estimators

Abstract linear estimation concerns the estimation of an abstract parameter that depends on the underlying density via a linear transformation. An important subclass is the class of inverse problems where this transformation is naturally described as the inverse of some bounded operator. Suitable preconditioning allows us to restrict ourselves to the inverse of some Hermitian operator, which does not remain restricted to the class of compact operators. A lower bound to the minimax risk is obtained for the class of all estimators satisfying a natural moment condition and certain submodels. To establish the bound we use the Bayesian van Trees inequality and systems of (pseudo) eigenvectors of the operator involved. We also briefly sketch a general construction method for estimators, based on a regularized inverse of the operator involved, and show that these estimators attain the asymptotic minimax rate in interesting examples.