Conditions are given for the attainment of the Hammersley-Chapman-Robbins bound for the variance of an unbiased estimator, in both regular and nonregular cases. Comparisons are made between this bound and the Bhattacharyya system of bounds for a wide class of distributions and parametric functions. Sufficient conditions are provided to determine when one bound is sharper than the other one.
"Comparison of Some Bounds in Estimation Theory." Ann. Statist. 4 (4) 755 - 765, July, 1976. https://doi.org/10.1214/aos/1176343547