  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. Cramer-Rao bound (CRB), a popular tool to
lower bound the variances of unbiased estimators, seems inapplicable in such
situations. In this paper, 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.
