New alternative derivations of Cramer-Rao bound for equality-constrained estimation

This paper presents new alternative derivations of the well-known constrained Cramer-Rao bound for estimating parameters that satisfy a set of deterministic and differentiable equality constraints. Specifically, for unbiased parameter estimation, the equality constraints are treated as pseudo-measurements corrupted by independent zero-mean Gaussian noise with infinitely small covariance. In this way, the desired constrained Cramer-Rao bound can be established via utilising the property that the Fisher information matrices for independent measurements are additive. This paper also provides a new simple way for deriving the constrained Cramer-Rao bound when the parameter estimate has a specified bias through exploring that the estimation error lies in the null space of the gradient matrix of the constraints and invoking the Cauchy-Schwarz inequality.

Automated summary

This paper introduces new alternative derivations for estimating the constrained Cramer-Rao bound of parameters subject to a set of deterministic and differentiable equality constraints. By treating equality constraints as pseudo-measurements corrupted by independent Gaussian noise, the desired bound can be established by utilizing the additive property of Fisher information matrices for independent measurements. It also provides a simple approach to deriving the constrained Cramer-Rao bound with specified bias in parameter estimates by exploring the null space of the gradient matrix of constraints and invoking the Cauchy-Schwarz inequality.