Optimal Two-Choice Stopping on an Exponential Sequence

Let X1 X2 Xn be independent and identically distributed with distribution function F . A statistician may choose two X values from the sequence by means of two stopping rules t1 t2, with the goal of maximizing E Xt1 ∨ Xt2 . We describe the optimal stopping rules and the asymptotic behavior of the optimal expected stopping values, V 2 n , as n → , when F is the exponential distribution. Specifically, we show that limn→ n 1− F V 2 n = 1− e−1, and conjecture that this same limit obtains for any F in the (Type I) domain of attraction of exp −e−x .