Second-Order Asymptotically Optimal Universal Outlier Hypothesis Testing

We revisit the universal outlier hypothesis testing (Li emph{et al.}, TIT 2014) and derive fundamental limits for the optimal test. In outlying hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated according to anomalous distributions. The nominal and anomalous distributions are emph{unknown}. We study the tradeoff among the probabilities of misclassification error, false alarm and false reject for tests that satisfy weak conditions on the rate of decrease of these error probabilities as a function of sequence length. Specifically, we propose a threshold-based universal test that ensures exponential decay of misclassification error and false alarm probabilities. We study two constraints on the false reject probabilities, one is that it be a non-vanishing constant and the other is that it have an exponential decay rate. For both cases, we characterize bounds on the false reject probability, as a function of the threshold, for each pair of nominal and anomalous distributions and demonstrate the optimality of our test in the generalized Neyman-Pearson sense. We first consider the case of at most one outlier and then generalize our results to the case of multiple outliers where the number of outliers is unknown and each outlier can follow a different anomalous distribution.