Second-order optimality of randomized estimation and test procedures

Let P([Theta], [tau]) || , [theta] [set membership, variant] [Theta] [subset of] , [tau] [set membership, variant] T [subset of] p denote a family of probability measures, where [tau] denotes the vector of nuisance parameters. Starting from randomized asymptotic maximum likelihood (as. m. l.) estimators for ([theta], [tau]) we construct randomized estimators which are asymptotically median unbiased up to o(n-1/2) resp. test procedures which are as. similar of level [alpha] + o(n-1/2) (for testing [theta] = [theta]0, [tau] [set membership, variant] T against one sided alternatives). The estimation procedures are second-order efficient in the class of estimators which are median unbiased up to o(n-1/2) and the test procedures are second-order efficient in the class of tests which are as. of level [alpha] + o(n-1/2). These results hold without any continuity condition on the family of probability measures.