Consider a regular $p$-dimensional exponential family such that either the distributions are concentrated on a lattice or they have a component whose $k$-fold convolution has a bounded density with respect to Lebesgue measure. Then, if a parametric function has an unbiased estimator, the minimum variance unbiased estimators are asymptotically equivalent to the maximum likelihood estimators; and, hence, are asymptotically efficient. Examples are given to show that a condition like the above is needed to obtain the asymptotic equivalence.
"Asymptotic Efficiency of Minimum Variance Unbiased Estimators." Ann. Statist. 5 (3) 522 - 529, May, 1977. https://doi.org/10.1214/aos/1176343849