More higher-order e ciency: Concentration probability

Based on concentration probability of estimators about a true parameter, third-order asymptotic e ciency of the rst-order bias-adjusted MLE within the class of rst-order bias-adjusted estimators has been well established in a variety of probability models. In this paper we consider the class of second-order bias-adjusted Fisher consistent estimators of a structural parameter vector on the basis of an i.i.d. sample drawn from a curved exponential-type distribution, and study the asymptotic concentration probability, about a true parameter vector, of these estimators up to the fth-order. In particular, (i) we show that third-order e cient estimators are always fourth-order e cient; (ii) a necessary and su cient condition for fth-order e ciency is provided; and nally (iii) the MLE is shown to be fth-order e cient.