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 R atio 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 ob servations is the SPRT, where this expectation is taken jointly with r espect to both data and prior over the two hypotheses. Third, the leve l a test needing the fewest conditional-expected number of observation s 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-t ime manner. In this paper, we relax some of the conditions and show th at 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 sequ ential procedures that possess the first two types of optimality. By r elaxing the one-at-a-time sampling restriction, we obtain optimal (in the first two senses) variable-sample-size-sequential probability rati o tests.