Lower bounds for parametric estimation with constraints
- 1 November 1990
- journal article
- Published by Institute of Electrical and Electronics Engineers (IEEE) in IEEE Transactions on Information Theory
- Vol. 36 (6), 1285-1301
- https://doi.org/10.1109/18.59929
Abstract
A Chapman-Robbins form of the Barankin bound is used to derive a multiparameter Cramer-Rao (CR) type lower bound on estimator error covariance when the parameter theta in R/sup n/ is constrained to lie in a subset of the parameter space. A simple form for the constrained CR bound is obtained when the constraint set Theta /sub C/, can be expressed as a smooth functional inequality constraint. It is shown that the constrained CR bound is identical to the unconstrained CR bound at the regular points of Theta /sub C/, i.e. where no equality constraints are active. On the other hand, at those points theta in Theta /sub C/ where pure equality constraints are active the full-rank Fisher information matrix in the unconstrained CR bound must be replaced by a rank-reduced Fisher information matrix obtained as a projection of the full-rank Fisher matrix onto the tangent hyperplane of the full-rank Fisher matrix onto the tangent hyperplane of the constraint set at theta . A necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable, and examples for which the bound is achieved are presented. In addition to providing a useful generalization of the CR bound, the results permit analysis of the gain in achievable MSE performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization.<>Keywords
This publication has 19 references indexed in Scilit:
- Lower bounds on parametric estimators with constraintsPublished by Institute of Electrical and Electronics Engineers (IEEE) ,2003
- Matrix AnalysisPublished by Cambridge University Press (CUP) ,1985
- The use of linear constraints to reduce the variance of time of arrival difference estimates for source locationIEEE Transactions on Acoustics, Speech, and Signal Processing, 1984
- Information and Asymptotic Efficiency in Parametric-Nonparametric ModelsThe Annals of Statistics, 1983
- Estimating the Angles of Arrival of Multiple Plane WavesIEEE Transactions on Aerospace and Electronic Systems, 1983
- Contributions to a General Asymptotic Statistical TheoryLecture Notes in Statistics, 1982
- Statistical EstimationPublished by Springer Nature ,1981
- Barankin Bounds on Parameter EstimationIEEE Transactions on Information Theory, 1971
- Maximum-likelihood estimation in non-standard conditionsMathematical Proceedings of the Cambridge Philosophical Society, 1971
- Minimum Variance Estimation Without Regularity AssumptionsThe Annals of Mathematical Statistics, 1951