On the Lower Bound of the Exact Asymptotics for the Large-Deviation Probabilities of Statistical Estimates

The Hajek–Le Cam theorem of locally asymptotic minimax bound of the risk of statistical estimates is extended to the problem of lower bound of the exact asymptotics for the large-deviation probability of statistical estimates. The result obtained can be considered as the exact lower bound of efficiency for confidence estimation of a multivariate parameter, inasmuch as the estimated confidence level usually assumes small values.