An Interpretation of the Moore-Penrose Generalized Inverse of a Singular Fisher Information Matrix

It is proved that in a non-Bayesian parametric estimation problem, if the Fisher information matrix (FIM) is singular, unbiased estimators for the unknown parameter will not exist. Cramér-Rao bound (CRB), a popular tool to lower bound the variances of unbiased estimators, seems inapplicable in such situations. In this correspondence, we show that the Moore-Penrose generalized inverse of a singular FIM can be interpreted as the CRB corresponding to the minimum variance among all choices of minimum constraint functions. This result ensures the logical validity of applying the Moore-Penrose generalized inverse of an FIM as the covariance lower bound when the FIM is singular. Furthermore, the result can be applied as a performance bound on the joint design of constraint functions and unbiased estimators.