  We consider the classical sequential binary hypothesis testing problem in
which there are two hypotheses governed respectively by distributions $P_0$ and
$P_1$ and we would like to decide which hypothesis is true using a sequential
test. It is known from the work of Wald and Wolfowitz that as the expectation
of the length of the test grows, the optimal type-I and type-II error exponents
approach the relative entropies $D(P_1\|P_0)$ and $D(P_0\|P_1)$. We refine this
result by considering the optimal backoff---or second-order asymptotics---from
the corner point of the achievable exponent region $(D(P_1\|P_0),D(P_0\|P_1))$
under two different constraints on the length of the test (or the sample size).
First, we consider a probabilistic constraint in which the probability that the
length of test exceeds a prescribed integer $n$ is less than a certain
threshold $0<\varepsilon <1$. Second, the expectation of the sample size is
bounded by $n$. In both cases, and under mild conditions, the second-order
asymptotics is characterized exactly. Numerical examples are provided to
illustrate our results.
