The first and second order large deviation efficiency is discussed from a different viewpoint from the Bahadur efficiency depending on the lower bound for the tail probability of estimators based on the amount of Kullback-Leibler information. The first and second order lower bounds for the tail probability of weakly asymptotically median unbiased (AMU) estimators are directly derived from the saddlepoint approximation. For an exponential family of distributions, it is shown that the maximum likelihood estimator (MLE) is second order large deviation efficient in the sense that the MLE attains the second order lower bound. For the Cauchy distribution with a location parameter, its MLE is shown to be first order large deviation efficient, which implies that it is asymptotically Bahadur efficient. Next, for a sum of independent discrete random variables, its higher order large deviation approximation is discussed. Its numerical comparison with other approximations is given in the binomial case.