Sufficiency and Information Rate of Multistage Statistical Tests

摘要

In this memorandum some fundamental aspects of multi-stage tests of alternate statistical hypotheses are discussed. Section II is devoted to the formulation of the problem and the definition of the quantities of interest. Section III demonstrates certain fundamental equalities of the conditional distributions of the sample size which occur in Wald's sequential probability ratio test. These equalities, which to the authors' knowledge have not been noted before, imply that the terminal decision is a sufficient statistic for the estimation of the true hypothesis regardless of the terminal stage. In Section IV a further consequence of these equalities is demonstrated. Using information theoretic concepts, the rate of transmission of a statistical test is defined and a test procedure, constructed to satisfy these equalities, is shown to minimize this rate. The results in the memorandum provide an alternate approach to the study of the optimality of multi-stage tests of alternate statistical hypotheses and suggest a criterion for designing such tests based on the conditional distributions of the sample size rather than on the average risk. (Author)