(A) simple and improved approximation method for the characteristics of sequential tests

Abstract

Sequential hypothesis testing procedures differ from other statistical procedures in that the sample size is not fixed in advance. They provide a viable alternative to the more traditional fixed sample size procedures in many situations of interest. Since sequential procedures can, on average, reduce the number of observations required to reach a decision, they are particularly attractive in cases where the act of gathering data is expensive, time-consuming, or requires destructive testing. Wald(1947) developed the sequential probability ratio test(SPRT) for testing a simple hypothesis against a simple alternative. The SPRT has an optimum property for these two hypotheses, namely, given such a test there is no other test with at least as low probabilities of type I and type II errors and with smaller average sample number under either or both of two hypotheses. Wald gave approximation formulas for the sequential test procedure, its average sample number, and the power of the test. However, Wald``s approximations have been shown to be inaccurate when applied in practice. Thus many studies have been done to approximate the characteristics of the test. One of the major efforts among them is to approximate the excess over the boundaries used in the test. In this thesis the excess is approximated as a simple function of the parameter to be tested by using the condition of the test statistic immediately before the stopping time in normal and exponential cases. The use of the estimated excess shows good performances in estimating the operating characteristic function, the average sample number, the moment and the probability mass function of the sample number. It also make it possible to determine the boundary values which can give the error probabilities close to the desired ones. To check on accuracy of the proposed method, it is compared to standard numerical and approximate methods in normal and exponential cases. From the results for normal and exponential cas...