Dual Cones, Dual Norms, and Simultaneous Inference for Partially Ordered Means

Abstract Exact simultaneous one-sided confidence intervals for contrasts in m normal means are discussed. The set K of contrast vectors considered is of one of two forms: Either it is a cone whose coordinates are monotone 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 obtained are inverted from test statistics that may be used for testing the null hypothesis that the mean vector μ lies in the dual (or negative polar) cone of K, against the alternative that μ is not in the dual of K. Corresponding conservative two-sided intervals are also discussed. The structure of such cones is considered and results concerning dual norms for such cones are obtained. Illustrations for simple ordering, simple-tree, and umbrella orderings are discussed.