Following the essential steps of the proof of the Cramer-Rao inequality [1, 2] but avoiding the need to transform coordinates or to differentiate under integral signs, a lower bound for the variance of estimators is obtained which is (a) free from regularity assumptions and (b) at least equal to and in some cases greater than that given by the Cramer-Rao inequality. The inequality of this paper might also be obtained from Barankin's general result . Only the simplest case--that of unbiased estimation of a single real parameter--is considered here but the same idea can be applied to more general problems of estimation.
"Minimum Variance Estimation Without Regularity Assumptions." Ann. Math. Statist. 22 (4) 581 - 586, December, 1951. https://doi.org/10.1214/aoms/1177729548