Sequential Outlier Hypothesis Testing under Universality Constraints

TL;DR

This paper develops bounds and tests for sequential outlier hypothesis testing when both nominal and anomalous distributions are unknown, providing tight bounds, improved exponents, and practical generalizations for multiple outliers.

Contribution

It introduces tight exponent bounds for the case of one outlier, proposes a threshold-based test for multiple outliers, and analyzes the impact of unknown number of outliers on error exponents.

Findings

Achieves exact large deviations characterization for one outlier.

Proposes a universal sequential test with larger Bayesian exponent than fixed-length tests.

Shows a penalty in error exponents when the number of outliers is unknown.

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, Nitinawarat and Veeravalli (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, Nitinawarat and Veeravalli (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 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, Wei and Hero (TIT 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.