Binary Experiments, Minimax Tests and 2-Alternating Capacities

Abstract

The concept of Choquet's 2-alternating capacity is explored from the viewpoint of Le Cam's experiment theory. It is shown that there always exists a least informative binary experiment for two sets of probability measures generated by 2-alternating capacities. This result easily implies the Neyman-Pearson lemma for capacities. Moreover, its proof gives a new method of construction of minimax tests for problems in which hypotheses are generated by 2-alternating capacities. It is also proved that the existence of least informative binary experiments is sufficient for a set of probability measures to be generated by a 2-alternating capacity. This gives a new characterization of 2-alternating capacities, closely related to that of Huber and Strassen.