  We consider the sequential composite binary hypothesis testing problem in
which one of the hypotheses is governed by a single distribution while the
other is governed by a family of distributions whose parameters belong to a
known set $\Gamma$. We would like to design a test to decide which hypothesis
is in effect. Under the constraints that the probabilities that the length of
the test, a stopping time, exceeds $n$ are bounded by a certain threshold
$\epsilon$, we obtain certain fundamental limits on the asymptotic behavior of
the sequential test as $n$ tends to infinity. Assuming that $\Gamma$ is a
convex and compact set, we obtain the set of all first-order error exponents
for the problem. We also prove a strong converse. Additionally, we obtain the
set of second-order error exponents under the assumption that $\mathcal{X}$ is
a finite alphabet. In the proof of second-order asymptotics, a main technical
contribution is the derivation of a central limit-type result for a maximum of
an uncountable set of log-likelihood ratios under suitable conditions. This
result may be of independent interest. We also show that some important
statistical models satisfy the conditions.
