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Abstract
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In the present paper, we are interested in testing the order restriction of mean vectors
against all alternatives based on a sample from several p dimensional normal distributions.
Here we consider the case when the covariance matrices are completely unknown but com-
mon. We propose a test statistic and obtain the supremum of its upper tail probability
under the null hypothesis. A reformulation of the test statistic is also provided based on the
orthogonal projections on the closed convex cones to study its null distribution. Finally, a
Monte Carlo simulation is presented to estimate its critical values.
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