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We consider the problem of testing the null hypothesis that a multivariate normal mean vector is constrained to lie in a set of points satisfying multiple linear inequality constraints. Such testing problem can also be described as one subclass of problems of testing multivariate one-sided null hypotheses. Unlike problems of testing simple against multivariate one-sided alternative hypotheses, only a few problems of testing multivariate one-sided null hypotheses have been investigated by Liu and Berger (1995), Perlman and Wu (2004). Our testing problem is different than the problems of Liu and Berger (1995) and is a more general testing problems than those studied by Perlman and Wu (2004).
The null hypothesis of our interest is a polyhedral convex set whose boundary is a union of faces of vary dimensions. We proposed a new test consisting of two stages: pretest and posttest. In the pretest stage, a face is selected according to the value of PIC (pretest information criterion). In the posttest stage, the posttest rejects the null hypothesis if the likelihood ratio statistic greater than a constant, also called a critical value, which is different for each possible choice of pretest. The critical values are determined so that the level of the test is α. The testing procedure is constructed such that the test is pointwise asymptotic unbiased.
For calculating the critical values, we provide analytic formulas for two cases: (1) testing with bivariate means and (2) testing with multivariate means constrained in a positive orthant. But for a general polyhedral convex set, we utilized the computational geometry algorithm Cdd to obtain the collection of faces and proposed an annealing Markov Chain Monte Carlo simulation method to estimate the rejection probability of the test. The critical values are solved in the Monte Carlo simulation.