Bayes tests for linearly constrained parameters

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Generalized Bayes test for loop-ordered normal means is derived and it is shown that the minimum power of the test is almost the same as that of the likelihood ratio test but that the maximum power is fairly larger than the likelihood ratio test. The polyhedral convex cone formed by the loop-ordered alternatives has a set of pairs of two corner vectors which are most distant from the center, intersecting with the central vector of the cone with the equal angle. It is shown that uniform prior should be taken on the corner vectors located most far from the center. The generalized Bayes test having the uniform prior on the most nearest corner vectors from the center has poor global power and is not recommended, though it seems intuitively appealing without geometrical consideration and has simple expression. The same consideration yields Bayes tests for simple ordered normal means and tree ordered normal means. Thegeneralized Bayes tests are also derived by taking the limit of half-normal priors.Bayes test for testing the homogeneity of variances before and after the change point in a normal linear regression model is derived. For noninformative prior, posterior distribution of the ratio of variances is derived and the confidence interval is constructed. It is shown that test based on confidence interval is superior to the usual test based on the maximum likelihood estimates in that it guarantees the type I error. The renal transplant data is analyzed by this Bayes test and is found that the variances before and after the operation do not change significantly.