Universal Outlier Hypothesis Testing

TL;DR

This paper investigates universal methods for detecting outlier sequences among multiple data streams, establishing conditions for exponential consistency and characterizing error exponents when distributions are unknown but distinct.

Contribution

It introduces a universal outlier hypothesis test that is exponentially consistent under certain conditions and characterizes the achievable error exponents.

Findings

Generalized likelihood test is universally exponentially consistent in some settings.

No universally exponentially consistent test exists in all settings.

Error exponents are explicitly characterized for the universal test.

Abstract

Outlier hypothesis testing is studied in a universal setting. Multiple sequences of observations are collected, a small subset of which are outliers. A sequence is considered an outlier if the observations in that sequence are distributed according to an ``outlier'' distribution, distinct from the ``typical'' distribution governing the observations in all the other sequences. Nothing is known about the outlier and typical distributions except that they are distinct and have full supports. The goal is to design a universal test to best discern the outlier sequence(s). It is shown that the generalized likelihood test is universally exponentially consistent under various settings. The achievable error exponent is also characterized. In the other settings, it is also shown that there cannot exist any universally exponentially consistent test.