Second-Order Asymptotics of Sequential Hypothesis Testing

Abstract

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 . 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.