  We 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 alternate hypothesis
$Q$, and only $P$ is known. For this problem, a well-known test is the
Hoeffding test, which accepts $P$ if the Kullback-Leibler (KL) divergence
between the empirical distribution of $Z^n$ and $P$ is below some threshold. In
this paper, we consider Hoeffding-like tests, where the KL divergence is
replaced by other divergences, and characterize, for a large class of
divergences, the first and second-order terms of the type-II error for a fixed
type-I error. Since the considered class includes the KL divergence, we obtain
the second-order term of the Hoeffiding test as a special case.
