P.B. Stark. Joint work with Y. Benjamini and V. Madar

Equivariance, optimality, unbiasedness, multiplicity, and the duality between tests and confidence sets are among the topics Erich Lehmann studied and addressed so beautifully in his books "Theory of Point Estimation" and "Testing Statistical Hypotheses." I will present a new method that exploits the duality between tests and confidence sets--but sacrifices equivariance and unbiasedness--to construct simultaneous confidence intervals for the components of a multivariate mean that determine the signs of the parameters more frequently than standard translation-equivariant intervals do. When one or more estimated means are small, the new intervals sacrifice some length to avoid crossing zero. But when all the estimated means are large, the new intervals coincide with standard, equivariant, simultaneous confidence intervals, so there is no loss of precision. The improvement can be substantial. For example, if two means are to be estimated and the intervals are allowed to be at most 80% longer than standard intervals in the worst case, then when only one mean is small its sign is determined almost as well as by a one-sided test that ignores multiplicity and has a pre-specified direction. When both are small the sign is determined better than by two-sided tests that ignore multiplicity. The intervals are constructed by inverting level-$\alpha$ tests to form a $1-\alpha$ confidence set, then projecting that set onto the coordinate axes to get confidence intervals. The tests have hyperrectangular acceptance regions that minimize the maximum amount by which the acceptance region protrudes from the orthant that contains the hypothesized parameter value, subject to a constraint on the maximum side length of the hyperrectangle. These tests are biased in general and the resulting confidence sets are equivariant under permutations of the coordinates and sign changes, but not under translation.