Moments of Implicitly Defined Estimators (e.g. ML and MAP): Applications to Transmission Tomography

Abstract

Many estimators in signal processing problems are defined implicitly as the maximum of an objective function, such as maximum likelihood (ML) and maximum a posteriori (MAP) methods. Exact analytical expressions for the mean and variance of such estimators are usually unavailable, so investigators usually resort to numerical simulations. The paper describes approximate analytical expressions for the mean and variance of implicitly defined estimators. The expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation. The authors demonstrate the utility and accuracy of the approximations in a PET transmission computed tomography application with Poisson statistics. The approximations should be useful in a wide range of estimation problems.