Beat the Mean: Better the Average

We consider a sequential rule, where an item is chosen into the group, such as a university faculty member, only if his score is better than the average score of those already belonging to the group. We study four variables: The average score of the members of the group after k items have been selected, the time it takes (in terms of number of observed items) to assemble a group of k items, the average score of the group after n items have been observed, and the number of items kept after the first n items have been observed. We develop the relationships between these variables, and obtain their asymptotic behavior as k (respectively, n) tends to infinity. The assumption throughout is that the items are independent, identically distributed, with a continuous distribution. Though knowledge of this distribution is not needed to implement the selection rule, the asymptotic behavior does depend on the distribution. We study in some detail the Exponential, Pareto and Beta distributions. Generalizations of the "better than average" rule to the β better than average rules are also considered. These are rules where an item is admitted to the group only if its score is better than β times the present average of the group, where β > 0.