Cramer-Rao Bound for Constrained Parameter Estimation Using Lehmann-Unbiasedness

TL;DR

This paper introduces a new, less restrictive unbiasedness concept called C-unbiasedness for constrained parameter estimation, along with a derived Lehmann-unbiased CCRB (LU-CCRB) that provides a more accurate lower bound on estimation error in non-asymptotic scenarios.

Contribution

The paper proposes C-unbiasedness based on Lehmann-unbiasedness with weighted MSE, and derives the LU-CCRB, a more informative lower bound applicable in non-asymptotic constrained estimation.

Findings

LU-CCRB is a valid lower bound where CCRB is not

C-unbiasedness is less restrictive than CCRB unbiasedness

Simulations show LU-CCRB outperforms CCRB in non-asymptotic cases

Abstract

The constrained Cramer-Rao bound (CCRB) is a lower bound on the mean-squared-error (MSE) of estimators that satisfy some unbiasedness conditions. Although the CCRB unbiasedness conditions are satisfied asymptotically by the constrained maximum likelihood (CML) estimator, in the non-asymptotic region these conditions are usually too strict and the commonly-used estimators, such as the CML estimator, do not satisfy them. Therefore, the CCRB may not be a lower bound on the MSE matrix of such estimators. In this paper, we propose a new definition for unbiasedness under constraints, 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 to C-unbiasedness, a Cramer-Rao-type bound on the WMSE of C-unbiased estimators, denoted as Lehmann-unbiased CCRB (LU-CCRB), is derived. This bound is a scalar bound that depends on the chosen weighted combination of estimation errors. It is shown that C-unbiasedness is less restrictive than the CCRB unbiasedness conditions. Thus, the set of estimators that satisfy the CCRB unbiasedness conditions is a subset of the set of C-unbiased estimators and the proposed LU-CCRB may be an informative lower bound in cases where the corresponding CCRB is not. In the simulations, we examine linear and nonlinear estimation problems under nonlinear constraints in which the CML estimator is shown to be C-unbiased and the LU-CCRB is an informative lower bound on the WMSE, while the corresponding CCRB on the WMSE is not a lower bound and is not informative in the non-asymptotic region.