It is often stated that the variance of an unbiased estimator of a function of a real parameter can attain the Cramer-Rao lower bound only if the family of distributions is a one-parameter exponential family. A rigorous proof of this statement, subject to certain regularity conditions, has been given by Wijsman. However, in general, the statement is not true. Assuming a revised set of regularity conditions it is shown here that there exists a more general class of distributions for which the Cramer-Rao lower bound for the variance is attained for almost all or even all values of the parameter in an interval. The class reduces to the exponential class only by imposing a restriction requiring the absolute continuity in the parameter of a function involving the logarithm of the probability density.
"On the Attainment of the Cramer-Rao Lower Bound." Ann. Statist. 4 (5) 998 - 1002, September, 1976. https://doi.org/10.1214/aos/1176343599