On asymptotic tests of composite hypotheses in nonstandard conditions

Abstract

The relationship between optimal C(α) tests of composite hypotheses and tests based on maximum likelihood estimators is discussed. It it shown that the two procedures are asymptotically equivalent if the vector parameter specified by the null hypothesis is interior to an open set in parameter space. Then considering these same tests when the parameter is on the boundary of a closed parameter space, the asymptotic distributional form of the maximum likelihood estimators is established under the null hypothesis. The nonequiva-lence of optimal C(α) and maximum likelihood tests in this boundary situation is discussed.