Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints

Jiachun Pan, Yonglong Li, Vincent Y. F. Tan

Abstract: 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Gamma$</tex> . 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> are bounded by a certain threshold <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon$</tex> , we obtain certain fundamental limits on the asymptotic behavior of the sequential test as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> tends to infinity. Assuming that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Gamma$</tex> 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, under the assumption that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\Gamma$</tex> is a finite set, we obtain the set of second-order error exponents.