Bayes tests for linearly constrained parameters

本文导出了环序正态均值的广义Bayes检验，证明了该检验的最小功效与似然比检验的最小功效几乎相同，而最大功效却比似然比检验大得多。由环序交替构成的多面凸锥有一组最远离中心的两个角向量对，它们与锥的中心向量以相等的角度相交。它表明，应采取统一的先验角向量位于最远离中心。广义贝叶斯检验在距离中心最近的角点向量上具有一致先验，但全局功效较差，不推荐使用，尽管它在直观上看起来很吸引人，没有几何考虑，并且具有简单的表达式。同样的考虑产生简单有序正态均值和树有序正态均值的贝叶斯检验。利用半正态先验的极限导出了广义Bayes检验，并导出了正态线性回归模型变点前后方差齐性的Bayes检验。对于无信息先验，推导了方差比的后验分布，并构造了置信区间。结果表明，基于置信区间的检验比基于极大似然估计的检验具有上级的优越性，因为它保证了I型误差。对肾移植数据进行贝叶斯检验，发现手术前后的方差变化不显著。

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.