Abstract: | Let P(Θ, τ) 6 , θ ∈ Θ ? , τ ∈ T ? p denote a family of probability measures, where τ denotes the vector of nuisance parameters. Starting from randomized asymptotic maximum likelihood (as. m. l.) estimators for (θ, τ) we construct randomized estimators which are asymptotically median unbiased up to resp. test procedures which are as. similar of level (for testing θ = θ0, τ ∈ T against one sided alternatives). The estimation procedures are second-order efficient in the class of estimators which are median unbiased up to and the test procedures are second-order efficient in the class of tests which are as. of level . These results hold without any continuity condition on the family of probability measures. |