Are Super-efficient Estimators Super-powerful?

Abstract We consider inference problems in two parametric families where estimators which are super-efficient relative to the maximum likelihood estimator (mle) exist. We examine whether tests based on these super-efficient estimators perform better than the tests based on the mle's. The measures of test efficiency used are Pitman efficiency, Bayes risk efficiency, Bahadur and Hodges–Lehmann efficiencies.