On Universality and Training in Binary Hypothesis Testing

The classical binary hypothesis testing problem is revisited. We notice that when one of the hypotheses composite, there an inherent difficulty in defining optimality criterion both informative and well-justified. For simple normal location (that is, for mean multivariate Gaussians), we overcome as follows. In this exists a natural “hardness” order between parameters different error-probabilities curves (when parameter known) are either identical, or dominates other. can thus define minimax performance worst-case among which below some hardness level. Fortunately, universal test, sense it all levels simultaneously. Under also find optimal test composite with training data. THIS extends to wide class local asymptotic models, where approximation error probabilities additive. Since have asymptotically tests without data, quantify loss universality gain data these models.