Sequential Sampling: Two Decision Problems with Linear Losses for Binomial and Normal Random Variables

In this paper we provide numerical solutions to the following decision problem. There are two terminal decisions and the states of the world are indexed by a parameter p, 0 < p < 1. The losses (or utilities) are linear in p. It is possible to observe independent and identically distributed random variables x1, x2, .... The probability that any xi = 1 is p, if not xi = 0. The cost of observing any xi is a constant. There exists a prior distribution of p which belongs to the beta family. A decision may be made after any number (including zero) of the random variables have been observed either to make one of the two terminal decisions or to observe a further xi. What is the optimum decision function ? The reason for considering this problem is that there are many situations in science and technology where it is possible to observe whether (xi = 1) or not (xi = 0) items possess a certain attribute and where it is necessary to make one of two decisions whose outcomes depend on the proportion p of items possessing the attribute. Under certain conditions the best way to make the decision is to proceed sequentially, that is by looking at the items one at a time, and after each inspection deciding whether to examine further items or to make a terminal decision. After xl, ..., Xn have been observed, and writing r = Exi, the likelihood is proportional to pr(l p)n-r. Consequently, it is simplest to use the beta prior distribution (see Raiffa & Schlaifer, 1961). The situation where the utilities are linear also recommends itself for first study. Possible applications of our results are to sampling inspection problems in industry, medical trials, sociological studies and other fields.