Universal Outlier Hypothesis Testing via Mean- and Median-Based Tests

2026-01-02

arXiv.org

AI Summary

This study addresses the problem of identifying anomalous sequences from a large collection of observed sequences when neither the typical nor the anomalous distributions are known, with both the total number of sequences and the number of anomalies growing as the sequence length increases. The work proposes two nonparametric detection methods based on the mean and the median: the mean-based approach is suited for sublinear growth of anomalies, while the median-based method is designed for the linear growth regime. The paper introduces the novel concept of the “typical error exponent” to characterize fundamental performance limits and, for the first time, employs median estimation in the setting where the anomaly proportion remains constant (i.e., linear growth). Theoretical analysis shows that both methods asymptotically achieve the error exponent of the optimal likelihood-ratio test under known distributions, with the median-based method attaining this optimality with probability approaching one.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Universal outlier hypothesis testing refers to a hypothesis testing problem where one observes a large number of length-$n$ sequences -- the majority of which are distributed according to the typical distribution $\pi$ and a small number are distributed according to the outlier distribution $\mu$ -- and one wishes to decide, which of these sequences are outliers without having knowledge of $\pi$ and $\mu$. In contrast to previous works, in this paper it is assumed that both the number of observation sequences and the number of outlier sequences grow with the sequence length. In this case, the typical distribution $\pi$ can be estimated by computing the mean over all observation sequences, provided that the number of outlier sequences is sublinear in the total number of sequences. It is demonstrated that, in this case, one can achieve the error exponent of the maximum likelihood test that has access to both $\pi$ and $\mu$. However, this mean-based test performs poorly when the number of outlier sequences is proportional to the total number of sequences. For this case, a median-based test is proposed that estimates $\pi$ as the median of all observation sequences. It is demonstrated that the median-based test achieves again the error exponent of the maximum likelihood test that has access to both $\pi$ and $\mu$, but only with probability approaching one. To formalize this case, the typical error exponent -- similar to the typical random coding exponent introduced in the context of random coding for channel coding -- is proposed.

Problem

• outlier detection
• hypothesis testing
• unknown distributions
• growing number of sequences
• typical error exponent

Innovation

• universal outlier detection
• mean-based test
• median-based test
• error exponent
• typical error exponent