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.