Multivariate isotonic regression theory plays a key role in the field of testing statistical hypotheses under order restriction for vector valued parameters. This kind of statistical hypothesis testing has been studied to some extent, for example, by Kulatunga and Sasabuchi (1984) when the covariance matrices are known and also Sasabuchi et al. (2003) and Sasabuchi (2007) when the covariance matrices are unknown but common. In the present paper, we are interested in a general testing for order restriction of mean vectors against all possible alternatives based on a random sample from several pdimensional normal populations when the unknown covariance matrices are common. In fact, this problem of testing is an extension of Bazyari and Chinipardaz’s (2012) problem. We propose a test statistic by likelihood ratio method based on orthogonal projections on the closed convex cones, study its upper tail probability under the null hypothesis and estimate its critical values for different significance levels by using Monte Carlo simulation. The problem of testing and obtained results is illustrated with a real example where this inference problem arises to evaluate the effect of Vinylidene fluoride on liver damage.
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