Characterizations of Complete Classes of Tests of Some Multiparametric Hypotheses, with Applications to Likelihood Ratio Tests

Abstract

For the problem of testing a simple hypothesis on a density function of the form $f_\theta(e) = \exp \{\psi_0(\theta) + \sum^k_1 \psi_i(\theta)t_i(e) + t_0(e)\}$, explicit characterizations are given of a minimal essentially complete class of tests, the minimal complete class, and the closure of the class of Bayes' solutions, under certain assumptions. Applications are made to discrete distributions of the above form and to some problems of testing composite hypotheses. The likelihood ratio tests of these hypotheses are characterized and shown to be admissible under certain assumptions.