Maximizing expected value with two stage stopping rules

Let X_n,…,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_n² 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_n² for a large class of F's belonging to the domain of attraction (for the maximum) D(G_II^?), where G_II^? (x) = exp(-x^-?)I(x > 0) with ? > 1. The results are compared with those for the asymptotic behavior of the classical one choice value sequence V_n¹, as well as with the "prophet value" sequence E(max{X_n,…,X₁}), and indicate that substantial improvement is obtained when given two chances to stop, rather than one.