Chernoff-Stein-Type Exponent in Testing Between Two Outlier Distributions

2026-08-06

AI Summary

This study addresses the hypothesis testing problem of detecting a single anomalous sequence among $2^{nR}$ sequences of length $n$, where the anomalous sequence follows distribution $Q_0$ under the null hypothesis and $Q_1$ under the alternative, while all other sequences are generated i.i.d. according to distribution $P$. Under a constraint on the type-I error probability, the work employs large deviation theory and information-theoretic techniques to establish, for the first time, the Chernoff–Stein exponent characterizing the optimal exponential decay rate of the type-II error probability in this setting. The result precisely quantifies the fundamental limit of detectability in terms of the trade-off between the code rate $R$ and the distinguishability among the underlying distributions.

Abstract

Among $2^{nR}$ length-$n$ random sequences, one sequence is an outlier whose index is random. Under hypothesis $\mathcal{H}_0$, the components of the outlier are independent and identically distributed (IID) according to $Q_0$, whereas under hypothesis $\mathcal{H}_1$ they are IID according to $Q_1$. The remaining $(2^{nR}-1)$ sequences are mutually independent, independent of the outlier, and IID according to $P$ under both hypotheses. Based on the observation of all $2^{nR}$ sequences, one wishes to decide between $\mathcal{H}_0$ and $\mathcal{H}_1$. Under the constraint that the decision error probability under $\mathcal{H}_0$ must be bounded away from $1$, we determine the fastest exponential decay rate of the decision error probability under $\mathcal{H}_1$.

Problem

• outlier detection
• hypothesis testing
• error exponent
• Chernoff-Stein lemma
• large deviations

Innovation

• Chernoff-Stein exponent
• outlier detection
• hypothesis testing
• error exponent
• large deviations