Second-Order Asymptotics of Sequential Hypothesis Testing

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 <; ε <; 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.