Decision Theory

15Stopping

In Chapter 14 we studied how to determine an optimal fixed sample size. In that discussion, the decision maker selects a sample size before seeing any of the data. Once the data are observed, the decision maker then makes a terminal decision, typically an inference. Here, we will consider the case in which the decision maker can make observations one at a time. At each time, the decision maker can either stop sampling (and choose a terminal decision), or continue sampling. The general technique for solving this sequential problem is dynamic programming, presented in general terms in Chapter 12.

In Section 15.1, we start with a little history of the origin of sequential sampling methods. Sequential designs were originally investigated because they can potentially reduce the expected sample size (Wald 1945). Our revisitation of Wald’s sequential stopping theory is almost entirely Bayesian and can be traced back to Arrow et al. (1949). In their paper, they address some limitations of Wald’s proof of the optimality of the sequential probability ratio test, derive a Bayesian solution, and also introduce a novel backward induction method that was the basis for dynamic programming. We illustrate the gains that can be achieved with sequential sampling using a simple example in Section 15.2. Next, we formalize the sequential framework for the optimal choice of the sample size in Sections 15.3.1, 15.3.2, and 15.3.3. In Section 15.4, we present an example of optimal stopping using ...