Asymptotic efficiencies of truncated sequential tests

Truncation of a sequential test with constant boundaries is considered for the problem of testing a location hypothesis: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x- \theta_{0})</tex> versus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x- \theta_{1})</tex> . A test design procedure is developed by using bounds for the error probabilities under the hypothesis and alternative. By viewing the truncated sequential test as a mixture of a sequential probability ratio test and a fixed sample size test, its boundaries and truncation point can be obtained once the degree of mixture is specified. Asymptotically correct approximations for the operating characteristic function and the average sample number function of the resulting test are derived. Numerical results show that an appropriately designed truncated sequential test performs favorably as compared to both the fixed sample size test and the sequential probability ratio test with the same error probabilities. The average sample number function of the truncated test is uniformly smaller than that of the fixed sample size test, and the truncated test maintains average sample sizes under the hypothesis and the alternative that are close to those optimum values achieved by Wald's sequential probability ratio test. Moreover, the truncated test is more favorable than the sequential probability ratio test in the sense that is has smaller average sample size when the actual location parameter is between <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta_{0}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta_{1}</tex> . This behavior becomes more pronounced as the error probabilities become smaller, implying that the truncated sequential test becomes more favorable as the error probabilities become smaller.