TY - UNPB
T1 - New Cramer-Rao-Type Bound for Constrained Parameter Estimation.
AU - Nitzan, Eyal
AU - Routtenberg, Tirza
AU - Tabrikian, Joseph
N1 - DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.
PY - 2018/11
Y1 - 2018/11
N2 - Non-Bayesian parameter estimation under parametric constraints is encountered in numerous applications in signal processing, communications, and control. Mean-squared-error (MSE) lower bounds are widely used as performance benchmarks and for system design. The well-known constrained Cram´er-Rao bound (CCRB) is a lower bound on the MSE of estimators that satisfy some unbiasedness conditions. In many constrained estimation problems, these unbiasedness conditions are too strict and popular estimators, such as the constrained maximum likelihood estimator, do not satisfy them. In addition, MSE performance can be uniformly improved by implementing estimators that do not satisfy these conditions. As a result, the CCRB is not a valid bound on the MSE of such estimators. In this paper, we propose a new definition for unbiasedness in constrained settings, denoted by C-unbiasedness, which is based on using Lehmann-unbiasedness with a weighted MSE (WMSE) risk and taking into account the parametric constraints. In addition, a Cram ´er-Rao-type bound on the WMSE of C-unbiased estimators, denoted as Lehmann-unbiased CCRB (LU-CCRB), is derived. It is shown that in general, C-unbiasedness is less restrictive than the CCRB unbiasedness conditions. Thus, the LU-CCRB is valid for a larger set of estimators than the CCRB and C-unbiased estimators with lower WMSE than the corresponding CCRB may exist. In the simulations, we examine linear and nonlinear estimation problems under nonlinear parametric constraints in which the constrained maximum likelihood estimator is shown to be C-unbiased and the LU-CCRB is an informative bound on its WMSE, while the corresponding CCRB is not valid.
AB - Non-Bayesian parameter estimation under parametric constraints is encountered in numerous applications in signal processing, communications, and control. Mean-squared-error (MSE) lower bounds are widely used as performance benchmarks and for system design. The well-known constrained Cram´er-Rao bound (CCRB) is a lower bound on the MSE of estimators that satisfy some unbiasedness conditions. In many constrained estimation problems, these unbiasedness conditions are too strict and popular estimators, such as the constrained maximum likelihood estimator, do not satisfy them. In addition, MSE performance can be uniformly improved by implementing estimators that do not satisfy these conditions. As a result, the CCRB is not a valid bound on the MSE of such estimators. In this paper, we propose a new definition for unbiasedness in constrained settings, denoted by C-unbiasedness, which is based on using Lehmann-unbiasedness with a weighted MSE (WMSE) risk and taking into account the parametric constraints. In addition, a Cram ´er-Rao-type bound on the WMSE of C-unbiased estimators, denoted as Lehmann-unbiased CCRB (LU-CCRB), is derived. It is shown that in general, C-unbiasedness is less restrictive than the CCRB unbiasedness conditions. Thus, the LU-CCRB is valid for a larger set of estimators than the CCRB and C-unbiased estimators with lower WMSE than the corresponding CCRB may exist. In the simulations, we examine linear and nonlinear estimation problems under nonlinear parametric constraints in which the constrained maximum likelihood estimator is shown to be C-unbiased and the LU-CCRB is an informative bound on its WMSE, while the corresponding CCRB is not valid.
U2 - https://doi.org/10.48550/arXiv.1802.02384
DO - https://doi.org/10.48550/arXiv.1802.02384
M3 - Preprint
BT - New Cramer-Rao-Type Bound for Constrained Parameter Estimation.
ER -