Chernoff-Stein-Type Exponent in Testing Between Two Outlier 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 $\mathcal{H}_1$.