  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.
