Directional tests for one-sided alternatives in multivariate models

Consider one-sided testing problems for a multivariate exponential family model. Through conditioning or other considerations, the problem oftentimes reduces to testing a null hypothesis that the natural parameter is a zero vector against the alternative that the natural parameter lies in a closed convex cone $\mathscr{C}$. The problems include testing homogeneity of parameters, testing independence in contingency tables, testing stochastic ordering of distributions and many others. A test methodology is developed that directionalizes the usual test procedures such as likelihood ratio, chi square, Fisher, and so on. The methodology can be applied to families of tests where the family is indexed by a size parameter so as to enable nonrandomized testing by $p$-values. For discrete models, a refined family of tests provides a refined grid for better testing by $p$-values. The tests have essential monotonicity properties that are required for admissibility and for desirable power properties. Two examples are given.