P3 approach to intersection-union testing of hypotheses

Recent applications of statistics often lead one to encounter testing problems where the original hypothesis of interest comprises the union of several sub-hypothesis. In the framework of such intersection-union (IU) testing of hypothesis, in contrast to the usual union-intersection (UI) framework, a sub-hypothesis therein may specify a parameter or a function of some of the parameters of the underlying distribution. The parameters may even be constrained to lie on the boundary of their parameter spaces. Even large-sample tests such as the usual likelihood ratio. Lagrangian multiplier or the Wald's tests then do not apply as their usual asymptotic distribution theory remain no longer be valid. An approach based on a pivotal parametric product P3 is enhanced here. It is shown that this approach often leads to appealing simple and elegant test statistics. The exact cut-off points and the power values can be computed by judicious use of numerical packages. L-optimality of such a test for the mixture problem is established. For multivariate multiparameter testing problems it is shown that such an approach leads to UI-IU tests. Construction of such tests are exemplified through several real-life problems as in, e.g., testing for interval specifications in acceptance sampling, for generalized variance of structured correlation matrices in generalized canonical variable, for agreement in method comparison studies, for no contamination in multiparameter multivariate mixture models, etc. It is demonstrated for a real-life data set in an acceptance sampling problem that the proposed class of P3 tests includes the intuitive one existing in the literature.