Second-Order Asymptotics of Sequential Hypothesis Testing
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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 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 backoff, also coined second-order asymptotics, is characterized precisely. Examples are provided to illustrate our results.
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