Nonequivariant Simultaneous Confidence Intervals Less Likely to Contain Zero

Abstract

Abstract We present a procedure for finding simultaneous confidence intervals for the expectations μ = (μ j ) n j=1 of a set of independent random variables, identically distributed up to their location parameters, that yields intervals less likely to contain zero than the standard simultaneous confidence intervals for many μ ≠ 0. The procedure is defined implicitly by inverting a nonequivariant hypothesis test with a hyperrectangular acceptance region whose orientation depends on the unsigned ranks of the components of μ, then projecting the convex hull of the resulting confidence region onto the coordinate axes. The projection to obtain simultaneous confidence intervals implicitly involves solving n! sets of linear inequalities in n variables, but the optima are attained among a set of at most n 2 such sets and can be found by a simple algorithm. The procedure also works when the statistics are exchangeable but not independent and can be extended to cases where the inference is based on statistics for μ...