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We consider estimation for a multivariate location family. Between all confidence regions with volume less than a fixed value  L , we find the equivariant confidence region with the biggest coverage probability. This region maximizes the infimum of the coverage probability over all confidence regions with volume less than  L . As an application, we find an estimator of a location parameter with the property that minimizes the supremum of the probability that the error of the estimation exceeds a fixed constant. We also find a confidence region and an estimator having the previous properties, but based on the maximum likelihood estimator. In the one dimensional case, we find the Bahadur slope of the two obtained estimators. We show that except for certain families of distributions, the estimator based on the whole sample is superior to the estimator based upon the maximum likelihood estimator. Hence, we get that maximum likelihood estimators are not asymptotically sufficient.

Minimax estimators of the coverage probability of the impermissible error for a location family