Sequential Outlier Hypothesis Testing Under Universality Constraints

Abstract

We revisit sequential outlier hypothesis testing and derive bounds on achievable exponents when both the nominal and anomalous distributions are unknown. The task of outlier hypothesis testing is to identify the set of outliers that are generated from an anomalous distribution among all observed sequences where the rest majority are generated from a nominal distribution. In the sequential setting, one obtains a symbol from each sequence per unit time until a reliable decision could be made. For the case with exactly one outlier, our exponent bounds are tight, providing exact large deviations characterization of sequential tests and strengthening a previous result of Li et al. (2017). In particular, the average sample size of our sequential test is bounded universally under any pair of nominal and anomalous distributions and our sequential test achieves larger Bayesian exponent than the fixed-length test, which could not be guaranteed by the sequential test of Li et al. (2017). For the case with at most one outlier, we propose a threshold-based test that has bounded expected stopping time under mild conditions and we bound the exponential decay rate of error probabilities, a.k.a., error exponents, under each non-null hypothesis and the null hypothesis. Our sequential test resolves the tradeoff among the exponential decay rates of misclassification, false reject and false alarm probabilities for the fixed-length test of Zhou et al. (2022). Finally, with a further step towards practical applications, we generalize our results to the cases of multiple outliers and show that there is a penalty in the error exponents when the number of outliers is unknown.