On Partitioning a Sample with Binary-Type Questions in Lieu of Collecting Observations

Abstract The problem is to search for the t largest observations in a random sample of size n by asking binary-type questions of the people (or items) in the sample without collecting any exact data whatever. The unordered and ordered cases are both considered; in the latter case the complete ranking is of special interest. Two different criteria of optimality are considered: (a) to minimize the expected number of questions required and (b) to maximize the probability of terminating the search in at most r questions for specified r. Optimal procedures are found and compared; in some sense the solutions for these two criteria are close to each other. The analysis is nonparametric in the sense that it holds for any underlying sampling distribution, but the actual optimal procedures depend on the specified distribution. In the above, we count (the cost of) a question as one regardless of the number of people addressed; other models in which the cost depends on the number of people are considered only briefly...