The resolution of binary hypothesis testing

We consider the asymptotics of hypothesis testing between two memoryless distributions. While traditional analysis concentrates on the exponential regime, we state a resolution question, where error probabilities are held fixed and the distributions grow closer as the blocklength grows. We define an asymptotic resolution tradeoff, and evaluate it for the optimal rule, the likelihood ratio test (LRT). Further, we analyze the loss when one of the distributions is unknown. Unlike the exponential setting, universality has a cost in terms of resolution. We define an appropriate sense of optimality and show that the widely used generalized LRT (GLRT) is indeed asymptotically optimal in that sense. Thus we derive a resolution tradeoff for this setting, and quantify the cost of universality. Although the asymptotics of the LRT and GLRT are known in the statistical literature, this work allows to state them within an information-theoretic framework, reminiscent of finite-blocklength analysis in communications.