Hypothesis tests for normal means constrained by linear inequalities

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Abstract
Likelihood ratio tests for a set of k normal means are studied when both the null and alternative hypotheses impose homogeneous linear inequality constraints on the means. Examples of such constraints include hypotheses that the means are equal, that they are nonnegative, that they are linear, or that they are monotone. Several tests involving such constraints have appeared recently, and here we present theory that unifies and extends many of these results. Our approach makes use of the properties of polyhedral cones. We define conditions under which the exact distribution of the likelihood ratio statistic is available, and show that this distribution is of the chi-bar-square form. When the exact distribution is unavailable, it may still be possible to find a stochastic ordering of the test statistic, and thereby derive a conservative test. The asymptotic distribution of the test statistic is also studied. Our results do not require that estimates of the k means be independent, provided that their covariance structure is known up to a constant