Some comments on the Cramér-Rao bound when the information matrix is singular

The covariance between the estimator and the score must be a linear transformation of the Fisher information matrix. It has been shown that this is a necessary and sufficient condition for the existence of a finite Cramér-Rao bound. In this note we show that this condition is unnecessary to obtain the above bound if the matrix of the linear transformation is limited in norm. A unified approach based on the matrix of minimum norm enables us to obtain a meaningful Cramér-Rao bound with the regular inverse replaced by the Moore-Penrose pseudoinverse as suggested by several authors. In this case the lower bound is achieved if and only if the estimator is a (minimum norm) linear transformation of the score.