We present a procedure for finding simultaneous confidence intervals for the expectations μ = (μj)nj=1of 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 solvingn! sets of linear inequalities innvariables, but the optima are attained among a set of at mostn2such 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 that are independent but not necessarily identically distributed, provided that there are known functions of μ that are location parameters for the statistics. In the general case, however, it appears that alln! sets of linear inequalities must be examined to find the confidence intervals.