Asymptotic Expansions Related to Minimum Contrast Estimators

Abstract

This paper contains an Edgeworth-type expansion for the distribution of a minimum contrast estimator, and expansions suitable for the computation of critical regions of prescribed error (type one) as well as confidence intervals of prescribed confidence coefficient. Furthermore, it is shown that, for one-sided alternatives, the test based on the maximum likelihood estimator as well as the test based on the derivative of the log-likelihood function is uniformly most powerful up to a term of order $O(n^{-1})$. Finally, an estimator is proposed which is median unbiased up to an error of order $O(n^{-1})$ and which is--within the class of all estimators with this property--maximally concentrated about the true parameter up to a term of order $O(n^{-1})$. The results of this paper refer to real parameters and to families of probability measures which are "continuous" in some appropriate sense (which excludes the common discrete distributions).