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 typeI and type-II error exponents approach the relative entropies D(P-1 parallel to P-0) and D(P-0 parallel to P-1). We refine this result by considering the optimal backoff from the corner point of the achievable exponent region (D(P-1 parallel to P-0),D(P-0 parallel to P-1)) under the expectation constraint on the length of the test (or the sample size). We consider the expectation constraint in which the expectation of the sample size is bounded by n, and under mild conditions, characterize the backoff, also coined second-order asymptotics, precisely. Examples are provided to illustrate our results.