Two-stage test with possibility of rejection of decision

The multiple statistical hypotheses two-stage testing with possibility of rejecting of decision to make choice between hypotheses concerning the pair of groups of probability distributions is considered such that in the first stage one group of distributions is distinguished and then in the second stage, the true distribution is denoted between mentioned group of probability distributions. Description of characteristics of logarithmically asymptotically optimal (LAO) hypotheses testing with possibility of decision rejection and the matrix of optimal asymptotically interdependencies of all pairs of the error probability exponents or reliabilities are studied. The goal of research is to express the optimal functional relation between the reliabilities of LAO hypotheses testing by a pair of stages and to compare with the case of similar one-stage testing.