Maximizing Expected Value with Two Stage Stopping Rules

Let Xn, . . . , X1 be i.i.d. random variables with distribution function F and finite expectation. A statistician, knowing F , observes the X values sequentially and is given two chances to choose X’s using stopping rules. The statistician’s goal is to select a value of X as large as possible. Let V 2 n equal the expectation of the larger of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behavior of the sequence V 2 n for a large class of F ’s belonging to the domain of attraction (for the maximum) D(GII), where G α II(x) = exp(−x−α)I(x > 0) with α > 1. The results are compared with those for the asymptotic behavior of the classical one choice value sequence V 1 n , as well as with the “prophet value” sequence E(max{Xn, . . . , X1}), and indicate that substantial improvement is obtained when given two chances to stop, rather than one.