On reliability approach to multiple hypotheses testing and to identification of probability distributions of two stochastically related objects

This paper is devoted to study of characteristics of logarithmically asymptotically optimal (LAO) hypotheses testing and identification for a model consisting of two related objects. In considered general case of dependence it is supposed that L₁ possible probability distributions of states constitute the family of possible hypotheses for the first object and the second object is distributed according to one of L₁ × L₂ given conditional distributions depending on the distribution index and the current observed state of the first object. By one approach the matrix of interdependencies of all possible pairs of the error probability exponents (reliabilities) of asymptotically optimal tests of distributions of both objects is studied. The identification of the distributions of two objects gives an answer to the question whether r₁-th and r₂-th distributions occurred, or not on the first and the second objects, correspondingly. Interdependence of reliabilities for the LAO identification is determined for each pair of double hypotheses. By the second approach the optimal interdependencies of lower estimates of all possible pairs of corresponding reliabilities are found and lower estimates of reliabilities for the LAO identification are studied for all pairs of hypotheses.