A rigorous proof is given of the often stated fact that if the variance of an unbiased estimator of a function of a real parameter attains the Cramer-Rao lower bound then the family of distributions must be a one-parameter exponential family.
"On the Attainment of the Cramer-Rao Lower Bound." Ann. Statist. 1 (3) 538 - 542, May, 1973. https://doi.org/10.1214/aos/1176342419