On asymptotically efficient statistical inference for moderate deviation probabilities

We study the lower bounds of efficiency for the moderate deviation probabilities of tests and estimators. These bounds cover both the logarithmic and strong asymptotics. For the problems of hypothesis testing we propose a natural representation for the lower bounds of type I and type II error probabilities in terms of inverse function of the standard normal distribution. The lower bounds for the moderate deviation probabilities of estimators are deduced easily from the corresponding bounds in hypothesis testing.