  Consider a binary statistical hypothesis testing problem, where $n$
independent and identically distributed random variables $Z^n$ are either
distributed according to the null hypothesis $P$ or the alternative hypothesis
$Q$, and only $P$ is known. A well-known test that is suitable for this case is
the so-called Hoeffding test, which accepts $P$ if the Kullback-Leibler (KL)
divergence between the empirical distribution of $Z^n$ and $P$ is below some
threshold. This work characterizes the first and second-order terms of the
type-II error probability for a fixed type-I error probability for the
Hoeffding test as well as for divergence tests, where the KL divergence is
replaced by a general divergence. It is demonstrated that, irrespective of the
divergence, divergence tests achieve the first-order term of the Neyman-Pearson
test, which is the optimal test when both $P$ and $Q$ are known. In contrast,
the second-order term of divergence tests is strictly worse than that of the
Neyman-Pearson test. It is further demonstrated that divergence tests with an
invariant divergence achieve the same second-order term as the Hoeffding test,
but divergence tests with a non-invariant divergence may outperform the
Hoeffding test for some alternative hypotheses $Q$. Potentially, this behavior
could be exploited by a composite hypothesis test with partial knowledge of the
alternative hypothesis $Q$ by tailoring the divergence of the divergence test
to the set of possible alternative hypotheses.
