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A discussion of fundamental aspects of multistage tests of alternate statistical hypotheses. The Memorandum formulates the problem and the definition of quantities of interest, and demonstrates certain fundamental equalities of the conditional distributions of the sample size occurring in Wald's sequential probability ratio test. These equalities (to the author's knowledge, not noted before) imply that the terminal decision is a sufficient statistic for estimation of true hypothesis regardless of the terminal stage. The results provide an alternate approach to the study of optimality of multistage tests of alternate statistical hypotheses and suggest a criterion for designing tests based on the conditional distributions of the sample size rather than on the average risk.
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