Asymptotical Optimality of Sequential Universal Hypothesis Testing Based on the Method of Types

In this letter, we introduce a sequential version of universal hypothesis testing, where the goal of the test is not only to decide between the known null hypothesis and some other unknown alternative hypothesis, but also to use a stopping time to stop the test as soon as rejecting the null hypothesis. Motivated by the method of types in information theory, we establish the sequential test by Höeffding's universal test associated with a curved stopping boundary. It is proved that this sequential test uniformly asymptotically minimizes average sample size for any other sequential test.