Universal Sequential Outlier Hypothesis Testing

TL;DR

This paper develops a universal sequential hypothesis testing method to identify outlier sequences among multiple data streams, even when the outlier and typical distributions are arbitrarily close, with proven asymptotic optimality.

Contribution

It introduces a universal repeated significance test for sequential outlier detection that is asymptotically optimal and consistent under various unknown distribution settings.

Findings

The proposed test is universally consistent.

It is asymptotically optimal when the number of outliers is maximal and the typical distribution is known.

Performance converges to known distribution scenarios even when distributions are unknown.

Abstract

Universal outlier hypothesis testing is studied in a sequential setting. Multiple observation sequences are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are generated by an "outlier" distribution, distinct from a common "typical" distribution governing the majority of the sequences. Apart from being distinct, the outlier and typical distributions can be arbitrarily close. The goal is to design a universal test to best discern all the outlier sequences. A universal test with the flavor of the repeated significance test is proposed and its asymptotic performance is characterized under various universal settings. The proposed test is shown to be universally consistent. For the model with identical outliers, the test is shown to be asymptotically optimal universally when the number of outliers is the largest possible and with the typical distribution being known, and its asymptotic performance otherwise is also characterized. An extension of the findings to the model with multiple distinct outliers is also discussed. In all cases, it is shown that the asymptotic performance guarantees for the proposed test when neither the outlier nor typical distribution is known converge to those when the typical distribution is known.