The hypotheses testing problem under unknown parameter

A hypotheses testing problem for discrimination of two collections of distributions is considered. Each collection is defined by a density function depending on an unknown parameter which belongs to some finite dimensional compact. Two decision rules are proposed: the first one is based on the idea of using a part of the observations for decision making and the whole sample for the parameter estimation, and the second is based on the first idea and on averaging over the parameter set according to an artificial “posterior” distribution. It is proven that the proposed methods are asymptotically equivalent to the optimal Bayesian rule in the case where the parameter is known. The problem of hypotheses testing in the case where the parameter belongs to one of two given sets is also considered.