Asymptotics for Outlier Hypothesis Testing

We revisit the outlier hypothesis testing framework of Li \emph{et al.} (TIT 2014) and derive fundamental limits for the optimal test. In outlier 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 consider the case of multiple outliers where the number of outliers is unknown and each outlier can follow a different anomalous distribution. Under this setting, we study the tradeoff among the probabilities of misclassification error, false alarm and false reject. Specifically, we propose a threshold-based test that ensures exponential decay of misclassification error and false alarm probabilities. We study two constraints on the false reject probability, with one constraint being that it is a non-vanishing constant and the other being that it has an exponential decay rate. For both cases, we characterize bounds on the false reject probability, as a function of the threshold, for each tuple of nominal and anomalous distributions. Finally, we demonstrate the asymptotic optimality of our test under the generalized Neyman-Pearson criterion.