Exponentially Consistent Outlier Hypothesis Testing for Continuous Sequences

TL;DR

This paper develops exponentially consistent outlier hypothesis tests for continuous-valued sequences with unknown distributions, applicable to fixed-length, sequential, and two-phase testing scenarios, even with multiple outliers.

Contribution

It introduces distribution-free tests for continuous sequences, extending outlier detection methods to unknown distributions and multiple outliers, with proven exponential error decay.

Findings

Exponential decay of misclassification, false reject, false alarm probabilities.

Effective two-phase test balances complexity and performance.

Generalization to multiple unknown outliers.

Abstract

In outlier hypothesis testing, one aims to detect outlying sequences among a given set of sequences, where most sequences are generated i.i.d. from a nominal distribution while outlying sequences (outliers) are generated i.i.d. from a different anomalous distribution. Most existing studies focus on discrete-valued sequences, where each data sample takes values in a finite set. To account for practical scenarios where data sequences usually take real values, we study outlier hypothesis testing for continuous sequences when both the nominal and anomalous distributions are \emph{unknown}. Specifically, we propose distribution free tests and prove that the probabilities of misclassification error, false reject and false alarm decay exponentially fast for three different test designs: fixed-length test, sequential test, and two-phase test. In a fixed-length test, one fixes the sample size of each observed sequence; in a sequential test, one takes a sample sequentially from each sequence per unit time until a reliable decision can be made; in a two-phase test, one adapts the sample size from two different fixed values. Remarkably, the two-phase test achieves a good balance between test design complexity and theoretical performance. We first consider the case of at most one outlier, and then generalize our results to the case with multiple outliers where the number of outliers is unknown.