Performance bounds for constrained parameter estimation

In this paper, we propose a new class of lower bounds on the mean-squared error (MSE) in non-Bayesian constrained parameter estimation. The new class includes lower bounds on the MSE of any constrained-unbiased estimator, where the constrained-unbiasedness is defined for the first time using the Lehmann-unbiasedness. The proposed class of constrained lower bounds is derived by employing Cauchy-Schwarz inequality and it can be used to derive various bounds for constrained parameter estimation. For example, it is demonstrated that the constrained Cramér-Rao bound (CCRB) is a special case of the proposed class. In addition, the new constrained Hammersley-Chapman-Robbins bound (CHCRB) is derived by using this class. Finally, the CCRB and CHCRB are exemplified in the estimation of the eigenvalues of a structured covariance matrix subject to signal subspace constraints. It is shown that the proposed CHCRB is tighter than the CCRB at any signal-to-noise ratio.