The First- and Second-Order Large-Deviation Efficiency for an Exponential Family and Certain Curved Exponential Models

The first- and second-order large-deviation efficiency is discussed for an exponential family of distributions. The lower bound for the tail probability of asymptotically median unbiased estimators is directly derived up to the second order by use of the saddlepoint approximation. The maximum likelihood estimator (MLE) is also shown to be second-order large-deviation efficient in the sense that the MLE attains the lower bound. Further, in certain curved exponential models, the first- and second-order lower bounds are obtained, and the MLE is shown not to be first-order large-deviation efficient.