A Useful Class of Multistage Tests

This report considers a class of hypothesis tests in which the (null) hypothesis is rejected only after being rejected on the basis of each of two samples. The motivation for these procedures is that their average sample sizes under the null hypothesis will be very near that of a fixed-sample-size test using only the first sample, while their error-probability performance should be superior. The validity of these properties is verified numerically for the particular problem of location testing with normal errors by considering the power functions and average sample sizes of linear versions of these tests. Also, the Pitman asymptotic efficiencies of the proposed test are compared to those of fixed-sample-size tests and sequential probability ratio tests, indicating that the proposed tests are intermediate to these two alternatives. However, the complexity and other properties of the proposed tests are comparable to those of the fixed-sample-size test, making their use desirable for many applications.