Two-Stage Likelihood Ratio and Union–Intersection Tests for One-Sided Alternatives Multivariate Mean with Nuisance Dispersion Matrix

For a multinormal distribution with an unknown dispersion matrix, union-intersection (UI) tests for the mean against one-sided alternatives are considered. The null distribution of the UI test statistic is derived and its power monotonicity properties are studied. A Stain-type two-stage procedure is proposed to eliminate some of the inherent drawbacks of such tests. Some comparisons are also made with some recently proposed alternative conditional likelihood ratio tests.