Sequential Window Problems

Window problems have an important application when there is concern about the clustering of nonconforming units or when there is an early surge of information that allows us to terminate an experiment sooner than usual. Motivated by these ideas, in this paper we use probability generating functions to study the stopping rules and their effects related to waiting time random variables. The use of windows introduces many new challenging probability questions, but it also requires basic tables that are not now available; some of these tables are included in the present paper. Since waiting-time random variables and scan statistics are closely related, the exact results and the computational method in this paper could also be useful in scan statistics. In the final section we apply the sequential-window concept to the possibility of changing the usual rules for a chess tournament and show that they could lead to a smaller expected number of games and to a higher chance of finding the better player.