Bounds on the Maximum Sample Size of a Bayes Sequential Procedure

Abstract

Sufficient conditions for a Bayes sequential procedure to be truncated contained in Wald [12] and in Blackwell and Girshick [3] have been used by Sobel [11] and Mikhalevich [7] to show that under certain conditions a Bayes sequential test is truncated when the cost of observation does not depend on the unknown parameter. When this latter condition is not satisfied as illustrated by some sequential testing problems considered by Kiefer and Weiss [5] and Weiss [14], and by Anscombe [2] (in connection with ethical cost of medical trials) the sufficient condition referred to (see Corollary 3.1) is not always applicable. When it is applicable it generally provides an upper bound on the maximum sample size that a Bayes sequential procedure can take. For the purpose of computation of the Bayes procedure by backward induction it is desirable to know the exact stage of truncation as close as possible. Section 3 contains two theorems (Theorem 3.2 and 3.3) which provide better sufficient conditions for a Bayes sequential procedure to be truncated and simultaneously provide better bounds on the exact stage of truncation. Sufficient conditions are also provided for the upper bounds involved to be exact. Section 4 contains some results with the help of which the theorems of Section 3 may be applied to some special sequential testing problems in Section 5. Finally Section 6 contains proofs of the results of Sections 3 and 4.