DUAL CONES, DUAL NORMS, AND SIMULTANEOUS INFERENCE FOR PARTIALLY ORDERED MEANS

Exact simultaneous one-sided confidence intervals for contrasts in m n ormal means are discussed. The set K of Contrast vectors considered is of one of two forms: Either it is a cone whose coordinates are monoto ne for a partial ordering defined on the coordinate index set (1,...,m ) or it is the polar of such a cone. It is shown that the intervals ob tained are inverted from test statistics that may be used for testing the null hypothesis that the mean vector mu lies in the dual (or negat ive polar) cone of K, against the alternative that mu is not in the du al of K. Corresponding conservative two-sided intervals are also discu ssed. The structure of such cones is considered and results concerning dual norms for such cones are obtained. Illustrations for simple orde ring, simple-tree, and umbrella orderings are discussed.