Optimizing lengths of confidence intervals: fourth-order efficiency in location models

Under regularity conditions the maximum likelihood estimator of the location parameter in a location model is asymptotically efficient among translation equivariant estimators. Additional regularity conditions warrant third- and even fourth-order efficiency, in the sense that no translation equivariant estimator will yield shorter confidence intervals than the maximum likelihood estimator. Unlike the literature on this issue, the present article does not exclude estimators from competition by assuming them to have a stochastic expansion of certain type. This is achieved by establishing a bound on the performance of all translation equivariant estimators, namely via our so-called confidence interval inequality.