Chapter 6 - Stopping Time of Invariant Sequential Probability Ratio Testst

Abstract

Publisher Summary This chapter focuses on stopping time of invariant sequential probability ratio tests. Wald's (1945, 1947) sequential probability ratio test (SPRT) for testing a simple hypothesis against a simple alternative has been researched extensively, and many of its properties are well known. For instance, it possesses the Wald–Wolfowitz optimum property; power function (if there is an imbedding one-parameter family of distributions) and expected sample sizes can be obtained at least approximately. By contrast, very little is known in general about sequential tests of composite hypotheses, which may arise when nuisance parameters are present. One of the earliest examples is provided by the sequential t -test. No simple optimum property is known for these tests. However, there is one aspect of these tests on which, in the past 25 years, a considerable amount of work has been done. This concerns the random sample size N , also called stopping time. The most basic question is whether N is finite with probability one. A more refined question is how fast the tail probabilities in the distribution of N go to zero. Substantial progress has been made on these problems.