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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.
Stoica et al. (Mon,) studied this question.