In this paper the exponential rates of decrease and bounds on tail probabilities for consistent estimators are studied using large deviation methods. The asymptotic expansions of Bahadur bounds and exponential rates in the case of the maximum likelihood estimator are obtained. Based on these results we have obtained a result parallel to the Fisher-Rao-Efron result concerning second-order efficiency (see Efron, 1975). Our results also substantiate the geometric observation given by Efron (1975) that if the statistical curvature of the underlying distribution is small, then the maximum likelihood estimator is nearly optimal.
James C. Fu. "Large Sample Point Estimation: A Large Deviation Theory Approach." Ann. Statist. 10 (3) 762 - 771, September, 1982. https://doi.org/10.1214/aos/1176345869