On the Constrained CramÉr–Rao Bound With a Singular Fisher Information Matrix

The problem considered in this letter is to bound the performance of estimators of a deterministic parameter which satisfies given constraints. Specifically, previous work on the constrained Cramer-Rao bound (CRB) is generalized to include singular Fisher information matrices and biased estimators. A necessary and sufficient condition for the existence of a finite CRB is obtained. A closed form for an estimator achieving the CRB, if one exists, is also provided, as well as a necessary and sufficient condition for the existence of such an estimator. It is shown that biased estimators achieving the CRB can be constructed in situations for which no unbiased technique exists.