Parameter estimation problems with singular information matrices

The case of a singular Fisher information matrix (FIM) represents a significant complication for the theory of the Cramer-Rao lower bound (CRB) that is usually handled by resorting to the pseudoinverse of the Fisher matrix, We take a different approach in which the CRB is derived as the solution to an unconstrained quadratic maximization problem, which enables us to handle the singular case in a simple yet rigorous manner. When the Fisher matrix is singular, except under unusual circumstances, any estimator having the specified bias derivatives that figure in the CRB must have infinite variance.