On Bayes Tests for p ~ i Versus p > i :

Few if any obtain a measure of success in their chosen profession without the beneficent influence of others. The first author is no exception. His interest in optimal stopping problems was ignited as a student in 1965 when Professor Robbins was visiting the University of Minnesota. Perhaps a fitting way of expressing appreciation would be to use this occasion to introduce the second author, Xizhi Wu, a member of a new generation of students whose interest in optimal stopping has been kindled by those who have "made straight the way"-and in particular by the one we seek here to honor. The present subject is not new. Much of the topic has already been resolved by Moriguti and Robbins (1962). What is new is an attempt to approximate the optimal stopping rule analytically. There are several reasons e why one should want such an approximation: I.The exact rule cannot be obtained explicitly. In the present context, the required backward induction is not difficult, and it can easily be programmed on an office computer. But there are several continuous parameters associated with the problem, and one is faced with an infinity of different backward induction problems if all values of these parameters are to be considered. 2.A good analytic approximation can indicate how the various para.eters influence the optimal stopping rule. 3.A good analytic approximation can be used to describe asymptotic behavior. 4.Most importantly, a good analytic approximation suggests a stopping rule (in an obvious way) which is probably close to optiMal-a good SUbstitute for the harder-to-describe optimal stopping rule. A standard method of apprOXimation is to replace a discrete time optimal stopping problem by a continuous time free boundary problem. A clear introduction to the method has been given by Chernoff (1972,pages 88-100). A