Parameter Estimation Problems with Singular Information Matrices

The case off a singular Fisher information matrix 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. We conclude that singularity implies that no finite-variance unbiased estimate for a function of the parameters can exist except under highly restrictive - and generally artificial - conditions. This difficulty arises entirely from the singularity of the Fisher matrix, and it is not mitigated significantly by allowing the estimate to be biased.