Summary: In an exponential family of distributions, the problem of testing for homogeneity of a set of parameters against some restricted alternatives is considered. Incorporating Roy’s union-intersection (UI) principle, Rao’s efficient score test is extended to such restricted alternatives, and is shown to be asymptotically power-equivalent to the corresponding restricted likelihood ratio tests (for contiguous alternatives).
Asymptotic null distributions of such UI score test statistics and their rates of convergence are studied. Multivariate problems for general forms of restricted alternatives are considered in the same vein. It is also shown that the UI score test has Bahadur deficiency of order \(O(\log n)\).