Lower bounds on parametric estimators with constraints

It is demonstrated that implicit constraints of the form G(θ)=0 can be incorporated into the Cramer-Rao lower bound. Constraints on the estimator restrict the local movement of the estimator θ(X) under statistical fluctuation in the observation vector X. A modified Cramer-Rao bound can be obtained by incorporating these restrictions into the formulation of the unconstrained bound. Conversely, when the parameter is constrained, the observation vector X is only allowed to vary according to the probability distribution F_θ, where θ is confirmed to the constraint set. The Fisher information matrix for the case of a constrained parameter can be written as a projection of the unconstrained Fisher matrix onto a subspace defined by the constraint. Conditions under which an estimator achieves the lower bound for constrained estimation are derived, and an example of an estimator that achieves the bound for the linear Gaussian problem is presented