A comparison of sequential sampling procedures for selecting the better of two binomial populations

Abstract

SUMMARY A comparison is made between play-the-winner sampling and vector-at-a-time sampling for selecting the better of two binomial populations when the selection requirement is defined in terms of the ratio of the single trial probabilities of success rather than the difference. Thus the probability of correct selection is to be at least P* when the true ratio of probabilities of success 0 is at least 0*, where P* and 0* are prescribed. Termination rules based on the difference in the number of successes and inverse sampling are considered. It is shown that play-the-winner sampling is uniformly preferable to vector-at-a-time sampling for both termination rules, and that for 0* close to one the success lead termination rule leads to a smaller expected number of trials on the poorer treatment unless 0 is even closer to one. The problem of selecting the better of two binomial populations, i.e. the one with the higher probability of success p on a single trial, has usually been stated using the framework of ranking and selection procedures as follows. If P* and A* are preassigned constants, with < P* < 1 and 0 < A* < 1, the probability of a correct selection is to be at least P* when the true difference in the p-values is at least A*. Sobel & Weiss (1970) consider two sequential procedures for this selection problem in which sampling is terminated when the absolute difference in the number of successes for the two populations first reaches a pre- determined integer. We will refer to this as the success-lead rule for termination. In the first procedure, play-the-winner sampling is used in which sampling continues with the same population after each success and switches to the other population after each failure. In the second procedure, vector-at-a-time sampling is used in which two observations are taken at each stage, one from each population. In many contexts, it is desirable to have a small expected number of trials on the poorer population or treatment but using the success-lead rule it is not true uniformly in the parameter space for the p's that play-the- winner sampling is superior to vector-at-a-time sampling in this respect. In a second paper Sobel & Weiss (1971) consider the use of an inverse sampling rule when sampling is termi- nated when any one population has r successes. They show that in this case play-the-winner sampling is uniformly preferable to vector-at-a-time sampling in the sense that for the same probability requirement, the expected number of trials on the poorer treatment is always smaller. Truncated versions of the inverse sampling rule and the success-lead rule for play- the-winner sampling were proposed by Hoel (1972) and Fushimi (1973). Their stopping rules include the number of failures as well as the number of successes and mean that excessive sample sizes are avoided when the population probabilities of success are small.