Lower bounds for parametric estimation with constraints

A Chapman-Robbins form of the Barankin bound is used to derive a multiparameter Cramer-Rao (CR) type lower bound on estimator error covariance when the parameter θ∈Rⁿ is constrained to lie in a subset of the parameter space. A simple form for the constrained CR bound is obtained when the constraint set Θ_C, can be expressed as a smooth functional inequality constraint. It is shown that the constrained CR bound is identical to the unconstrained CR bound at the regular points of Θ_C, i.e. where no equality constraints are active. On the other hand, at those points θ∈Θ_C where pure equality constraints are active the full-rank Fisher information matrix in the unconstrained CR bound must be replaced by a rank-reduced Fisher information matrix obtained as a projection of the full-rank Fisher matrix onto the tangent hyperplane of the full-rank Fisher matrix onto the tangent hyperplane of the constraint set at θ. A necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable, and examples for which the bound is achieved are presented. In addition to providing a useful generalization of the CR bound, the results permit analysis of the gain in achievable MSE performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization