Asymptotics of Sequential Composite Hypothesis Testing Under Probabilistic Constraints

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 the alphabet of the observations
$\mathcal {X}$
is finite. 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.