Reliability of sequential hypothesis testing can be achieved by an almost-fixed-length test

The maximum type-I and type-II error exponents associated with the newly introduced almost-fixed-length hypothesis testing is characterized. In this class of tests, the decision-maker declares the true hypothesis almost always after collecting a fixed number of samples n; however in very rare cases with exponentially small probability the decision maker is allowed to collect another set of samples (no more than polynomial in n). This class of hypothesis tests are shown to bridge the gap between the classical hypothesis testing with a fixed sample size and the sequential hypothesis testing, and improve the trade-off between type-I and type-II error exponents.