ASYMPTOTIC RELATIVE EFFICIENCIES OF MULTISTAGE TESTS.

Asymptotic relative efficiencies of two-stage and multistage tests of a simple null hypothesis versus a simple location alternative are considered. Observations are taken in groups, not necessarily of the same size, and a test statistic is computed for comparison with thresholds. In a two-stage test, if the test statistic of the first group of observations exceeds an upper threshold, the alternative is accepted, and if it crosses a lower threshold, the null hypothesis is accepted. Otherwise, a second group of samples is taken and a second test is performed. Two different classes of two-stage tests are considered. One of them computes the test statistic in the second stage from observations in the second group alone, while the other uses both the first and second groups of observations. It is shown that these tests are less efficient than sequential probability ratio tests. With proper choices of parameters, the improvement over fixed-sample-size tests can be significant, especially when the error probabilities are small. However, the complexity of two-stage tests is comparable to that of fixed-sample-size tests, making their use desirable. The efficiency of k-stage tests, k less than 2, is also investigated with the conclusion that the behavior of a two-stage test can be enhanced by adding stages up to a point beyond which the effect of adding stages diminishes.