A Neyman-Pearson Proper Way to Universal Testing of Multiple Hypotheses Formed by Groups of Distributions

Abstract

The asymptotically optimal Neyman-Pearson procedures of detection for models characterized by M discrete probability distributions arranged into K, 2 ≤ K ≤ M groups considered as hypotheses are investigated. The sequence of tests based on a growing number of observations is logarithmically asymptotically optimal (LAO) when a certain part of the given error probability exponents (reliabilities) provides positives values for all other reliabilities. LAO tests sequences for some models of objects, including cases, when rejection of decision may be permitted, and when part, or all given error probabilities decrease subexponentially with an increase in the of number of experiments, are desined. For all reliabilities of such tests single-letter characterizations are obtained. A simple case with three distributions and two hypotheses is considered.