MEAN AND VARIANCE OF IMPLICITLY DEFINED BIASED-ESTIMATORS (SUCH AS PENALIZED MAXIMUM-LIKELIHOOD) - APPLICATIONS TO TOMOGRAPHY

Many estimators in signal processing problems are defined implicitly a s the maximum of some objective function, Examples of implicitly defin ed estimators include maximum likelihood, penalized likelihood, maximu m a posteriori, and nonlinear least squares estimation. For such estim ators, exact analytical expressions for the mean and variance are usua lly unavailable, Therefore, investigators usually resort to numerical simulations to examine properties of the mean and variance of such est imators, This paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters, We derive the approximations using the implicit function theorem, the Taylor expansion, and the chain rule, The expressions are defined solely in terms of the partial derivatives of whatever object ive function one uses for estimation, As illustrations, we demonstrate that the approximations work well in two tomographic imaging applicat ions with Poisson statistics, We also describe a ''plug-in'' approxima tion that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement, The approximations s hould be useful in a wide range of estimation problems.