The Cramer-Rao inequality provides, under certain regularity conditions, a lower bound for the variance of an estimator , . Various generalizations, extensions and improvements in the bound have been made, by Barankin , , Bhattacharyya , Chapman and Robbins , Fraser and Guttman , Kiefer , and Wolfowitz , among others. Further considerations of certain inequality properties of a measure of information, discussed by Kullback and Leibler , yields a greater lower bound for the information measure (formula (4.11)), and leads to a result which may be considered a generalization of the Cramer-Rao inequality, the latter following as a special case. The results are used to define discrimination efficiency and estimation efficiency at a point in parameter space.
S. Kullback. "Certain Inequalities in Information Theory and the Cramer-Rao Inequality." Ann. Math. Statist. 25 (4) 745 - 751, December, 1954. https://doi.org/10.1214/aoms/1177728660