On the Computation of the Sampling Characteristics of a General Class of Sequential Decision Problems

Abstract

The connection between the theory of random walks and Wald's theory [10], [11] of sequential probability ratio tests of hypotheses has been remarked by several authors. In particular, Kemperman [5] has exploited that connection to obtain integral equations for the determination of the decision probabilities and the expected sample size of a Wald sequential test. It is the purpose of the present paper (1) to generalize Kemperman's integral equations to apply to a fairly extensive class of sequential multiple decision problems, and (2) to indicate methods of obtaining practical results from such integral equations. Part I of the paper is purely theoretical. It presents the integral equations already mentioned and generalizes a method of obtaining upper and lower bounds for their solutions that seems to have been first published by Kemperman [5] and Snyder [7] simultaneously. In Part II the possibilities for application of the general theory are illustrated by a discussion of Wald's sequential tests for simple alternatives on the parameter of a distribution, under the hypothesis that a sufficient statistic for that parameter exists. In particular, it is shown that the Kemperman-Snyder method for obtaining bounds for the solutions of the integral equations may be used to obtain substantial improvements over the bounds given by Wald for the operating characteristics of the test for simple alternatives on the mean of a normal distribution. Methods of numerical analysis are indicated that might be useful in a well-equipped computing laboratory for further improvement of the bounds. It is clear from the results obtained here that the methods used, coupled with extensive numerical work, should yield definitive improvements over Wald's approximate methods for setting the decision boundaries and estimating the sample size moments for sequential tests. It is hoped that the decision rule adopted in Part I is sufficiently general that the theory will provide a useful tool in the design and study of multiple decision problems.