Optimal Two-Choice Stopping on an Exponential Sequence

摘要

Let X_1,X_2,... ,X_n 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 t_1,t_2, with the goal of maximizing E(X_(t_1) v X_(t_2)). We describe the optimal stopping rules and the asymptotic behavior of the optimal expected stopping values, V_n~2, as n → ∞, when F is the exponential distribution. Specifically, we show that lim_(n→∞)(1 — F(V_n~2)) = 1 — e~(-1), and conjecture that this same limit obtains for any F in the (Type Ⅰ) domain of attraction of exp(—e~(-x)).