Integrated Risk of Asymptotically Bayes Sequential Tests

Abstract

For general multiple-decision testing problems, and even two-decision problems involving more than two states of nature, how to construct sequential procedures which are optimal (e.g. minimax, Bayes, or even admissible) is an open question. In the absence of optimality results, many procedures have been proposed for problems in this category. Among these are the procedures studied in Wald and Sobel (1949), DonnellY. (1957), Anderson (1960), and Schwarz (1962), all of which are discussed in the introduction of the paper by Kiefer and Sacks (1963) along with investigations in sequential design of experiments (notably those of Chernoff (1959) and Albert (1961)) which can be regarded as considering, inter alia, the (non-design) sequential testing problem. The present investigation concerns certain procedures which are asymptotically Bayes as the cost per observation, c, approaches zero and are definable by a simple rule: continue sampling until the a posteriori risk of stopping is less than Qc (where Q is a fixed positive number), and choose a terminal decision having minimum a posteriori risk. This rule, with Q = 1, was first considered by Schwarz and was shown to be asymptotically Bayes, under mild assumptions, by Kiefer and Sacks (whose results easily extend to the case of arbitrary Q > 0). Given an a priori distribution, F, and cost per observation, c, we shall use δ_F( Qc) to denote the procedure defined by this rule and δ_F * (c) to denote a Bayes solution with respect to F and c. The result of Kiefer and Sacks, for Q = 1, states that rc(F, δF(c)),....., r_c(F, δ_F*(c)) as c ~ 0, where rc(F, δ) is the integrated risk of δ when F is the a priori distribution and c is the cost per observation. The principal aim of the present work is to construct upper bounds (valid for all c > 0) on the difference r_c(F, δF(Qc)) - rc(F, δF*(c)), so that one can determine values of c (or the probabilities of error) small enough to insure that simple asymptotically optimum procedures are reasonably efficient.