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

Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints