Abstract
Let be the set of real numbers, the set of Borel sets of , and a -finite nonnegative measure on . Let be an open real number interval (which may be infinite). Throughout we consider a Koopman-Darmois family of generalized probability density functions on the measure space . We consider one sided tests of the hypothesis against the alternative . In general, in this paper, will be a sequential procedure. Associated with is a stopping variable (mention of the dependence of on is usually omitted). means that sampling stopped after observations and a decision was made. In this context we consider to be an integer, and means that sampling does not stop. In the discussion of Section 1 we will assume that if and then , that is, sampling stops with probability one. The reason for the exclusion of will become apparent in Section 1. We will be concerned with two events, decide , and, decide . The main result of this paper may be stated as follows. Theorem 1. Suppose , and are as described above. Define and assume . Suppose and and Then If there is a generalized sequential probability ratio test with stopping variable such that for the test , for the test , For all tests , if then . In Section 1, (7) and (8), it is shown that . Consequently the relations (4) and (5) of Theorem 1 are not vacuous. We were led to formulate Theorem 1 by a problem of constructing bounded length confidence intervals. The relationship is explained in Section 2. The proof of Theorem 1 is given in Section 3.
Citation
R. H. Farrell. "Asymptotic Behavior of Expected Sample Size in Certain One Sided Tests." Ann. Math. Statist. 35 (1) 36 - 72, March, 1964. https://doi.org/10.1214/aoms/1177703731
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