NONEQUIVARIANT SIMULTANEOUS CONFIDENCE-INTERVALS LESS LIKELY TO CONTAIN ZERO

We present a procedure for finding simultaneous confidence intervals f or the expectations mu = (mu(j))(j=1)(n) of a set of independent rando m variables, identically distributed up to their location parameters, that yields intervals less likely to contain zero than the standard si multaneous confidence intervals for many mu not equal 0. The procedure is defined implicitly by inverting a nonequivariant hypothesis test w ith a hyperrectangular acceptance region whose orientation depends on the unsigned ranks of the components of mu, then projecting the convex hull of the resulting confidence region onto the coordinate axes. The projection to obtain simultaneous confidence intervals implicitly inv olves solving n! sets of linear inequalities in n variables, but the o ptima are attained among a set of at most n(2) such sets and can be fo und by a simple algorithm. The procedure also works when the statistic s are exchangeable but not independent and can be extended to cases wh ere the inference is based on statistics for mu that are independent b ut not necessarily identically distributed, provided that there are kn own functions of mu that are location parameters for the statistics. I n the general case, however, it appears that all n! sets of linear ine qualities must be examined to find the confidence intervals.