Properties of Bayes testing procedures in order restricted inference

Consider k-independent normal populations with unknown means. Test the null hypothesis that the vector of means lies in a linear subspace (For example, the null could be all parameters are equal.) The alternative is that the vector of means lies in a closed convex cone (but not a linear subspace) whose dual cone is orthogonal to the linear subspace. Cohen et al. (2000, J. Multivariate Anal. 72, 50-77) showed that for many such cones the likelihood ratio test lacked a practical monotonicity property. Its behavior in such cases may be cause for concern. In this paper, we show that many Bayes tests also lack the practical monotonicity property.