Second-Order Asymptotically Optimal Outlier Hypothesis Testing

TL;DR

This paper develops a fundamental limit analysis and proposes an optimal threshold-based test for outlier hypothesis testing with unknown distributions, balancing error probabilities in multiple sequence scenarios.

Contribution

It introduces a new threshold-based testing method that achieves optimal tradeoffs among error probabilities under generalized Neyman-Pearson criteria for unknown distributions.

Findings

Proposed a test ensuring exponential decay of misclassification and false alarm errors.

Derived bounds on false reject probability for different constraints.

Proved the optimality of the test under the generalized Neyman-Pearson criterion.

Abstract

We revisit the outlier hypothesis testing framework of Li \emph{et al.} (TIT 2014) and derive fundamental limits for the optimal test under the generalized Neyman-Pearson criterion. In outlier hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated from anomalous distributions. The nominal and anomalous distributions are \emph{unknown}. We study the tradeoff among the probabilities of misclassification error, false alarm and false reject for tests that satisfy weak conditions on the rate of decrease of these error probabilities as a function of sequence length. Specifically, we propose a threshold-based test that ensures exponential decay of misclassification error and false alarm probabilities. We study two constraints on the false reject probability, with one constraint being that it is a non-vanishing constant and the other being that it has an exponential decay rate. For both cases, we characterize bounds on the false reject probability, as a function of the threshold, for each pair of nominal and anomalous distributions and demonstrate the optimality of our test under the generalized Neyman-Pearson criterion. We first consider the case of at most one outlying sequence and then generalize our results to the case of multiple outlying sequences where the number of outlying sequences is unknown and each outlying sequence can follow a different anomalous distribution.