A comparison of the cost-efficiencies of the sequential, group-sequential, and variable-sample-size-sequential probability ratio tests

Wald and Wolfowitz (1948) have shown that the Sequential Probability Ratio Test (SPRT) for deciding between two simple hypotheses is, under very restrictive conditions, optimal in three attractive senses. First, it can be a Bayes-optimal rule. Second, of all level a tests having the same power, the test with the smallest joint-expected number of observations is the SPRT, where this expectation is taken jointly with respect to both data and prior over the two hypotheses. Third, the level a test needing the fewest conditional-expected number of observations is the SPRT, where this expectation is now taken with respect to the data conditional on either hypothesis being true. Principal among the strong restrictions is that sampling can proceed only in a one-at-a-time manner. In this paper, we relax some of the conditions and show that there are sequential procedures that strictly dominate the SPRT in all three senses. We conclude that the third type of optimality occurs rarely and that decision-makers are better served by Looking for sequential procedures that possess the first two types of optimality. By relaxing the one-at-a-time sampling restriction, we obtain optimal (in the first two senses) variable-sample-size-sequential probability ratio tests.