Conditionally least-informative distributions: the case of the location parameter

We extend a basic result of Huber's on least favorable distributions to the setting of conditional inference, using an approach based on the notion of log-Gâteaux differentiation and perturbed models. Whereas Huber considered intervals of fixed width for location parameters and their average coverage rates, we study error models having longest confidence intervals, conditional on the location configuration of the sample. Our version of the problem does not have a global solution, but one that changes from configuration to configuration. Asymptotically, the conditionally least-informative shape minimizes the conditional Fisher information. We characterize the asymptotic solution within Huber's contamination model.