Exponentially Consistent Low Complexity Tests for Outlier Hypothesis Testing with Distribution Uncertainty

We revisit the outlier hypothesis testing (OHT) problem of Li et al. (TIT 2024) and propose exponentially consistent tests when there is distribution uncertainty for both nominal samples and outliers. In original OHT, one is given a list of sequences, most of which are generated i.i.d. from a distribution called the nominal distribution while the rest are generated i.i.d. from another distribution named the anomalous distribution. The task of OHT is to identify outliers when both the nominal and anomalous distributions are unknown. Motivated by the study for classification with distribution uncertainty by Hsu and Wang (ISIT 2020), we consider OHT with distribution uncertainty, where each nominal sample is generated from a distribution centered around the unknown nominal distribution and each outlier is generated from a distribution centered around the unknown anomalous distribution. With a further step towards practical applications, in the spirit of Bu et al. (TSP 2019), we propose low-complexity tests when the number of outliers is known and unknown, and show that our proposed tests are exponentially consistent. Furthermore, we demonstrate that there is a penalty for not knowing the number of outliers in the error exponent when outliers exist. Our results strengthen Bu et al. in three aspects: i) our tests allow distribution uncertainty and reveal the impact of distribution uncertainty on the performance of low-complexity tests; ii) when the number of outliers is known and there is no distribution uncertainty, our test achieves the same asymptotic performance with lower complexity; and iii) when the number of outliers is unknown, we characterize the tradeoff among the three error probabilities, while two of these error probabilities were not analyzed by Bu et al. even when there is no distribution uncertainty. Finally, we illustrate our theoretical results using numerical examples.