Admissible Representation of Asymptotically Optimal Estimates

Abstract

A sequence of medians of posterior distributions is approximately median unbiased of order $o(n^{-1}) \operatorname{iff}$ the prior density is equal to the square root of Fisher's information function. It is shown that in this case the sequence of medians of posterior distributions is even an optimum sequence of estimates within the class of all estimator sequences being approximately median unbiased of order $o(n^{-1}).$ The result is proved by showing equivalence with an expansion of an optimum sequence given by Pfanzagl. In the case of a location parameter family the Bayesian representation is admissible.