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 as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analyti...

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Veröffentlicht in:IEEE transactions on image processing : a publication of the IEEE Signal Processing Society. - 1992. - 5(1996), 3 vom: 15., Seite 493-506
1. Verfasser: Fessler, J A (VerfasserIn)
Format: Aufsatz
Sprache:English
Veröffentlicht: 1996
Zugriff auf das übergeordnete Werk:IEEE transactions on image processing : a publication of the IEEE Signal Processing Society
Schlagworte:Journal Article
Beschreibung
Zusammenfassung:Many estimators in signal processing problems are defined implicitly as the maximum of some objective function. Examples of implicitly defined estimators include maximum likelihood, penalized likelihood, maximum a posteriori, and nonlinear least squares estimation. For such estimators, exact analytical expressions for the mean and variance are usually unavailable. Therefore, investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators. 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 objective function one uses for estimation. As illustrations, we demonstrate that the approximations work well in two tomographic imaging applications with Poisson statistics. We also describe a "plug-in" approximation that provides a remarkably accurate estimate of variability even from a single noisy Poisson sinogram measurement. The approximations should be useful in a wide range of estimation problems
Beschreibung:Date Completed 02.10.2012
Date Revised 16.03.2022
published: Print
Citation Status PubMed-not-MEDLINE
ISSN:1057-7149