Large Deviations for Outlier Hypothesis Testing with Distribution Uncertainty

The task of outlier hypothesis testing is to identify outliers from a set of observed sequences, where most sequences named nominal samples are generated from nominal distributions and the rest sequences called outliers are generated from anomalous distributions. Inspired by the study of binary classification with distribution mismatch by Hsu and Wang (ISIT 2020), we study outlier hypothesis testing with distribution uncertainty. Specifically, each nominal sequence is generated from a distribution close to a centered nominal distribution and each outlier is generated from a distribution close to a centered anomalous distribution, where the closeness is measured via L-norms. Both nominal and anomalous distributions are unknown. To solve the above problem, we propose a threshold-based test and characterize the performance of our test in both the Chernoff's and Stein's regimes. Furthermore, in the Stein's regime, we analyze the impact of distribution uncertainty on the asymptotic performance of our result and show that the dominant term is a function of the likelihood ratio between centered nominal and anomalous distributions.