A sharp bound for winning within a proportion of the maximum of a sequence

Abstract

This note considers a variation of the full-information secretary problem where the random variables to be observed are continuous and iid. Consider the independent sequence of continuous random variables $X_1,\ldots ,X_n$, $M_n:=\max \{X_1,\ldots ,X_n\}$, and the objective to select the maximum of the sequence. What is the maximum probability of “stopping at the maximum”? That is, what is the stopping time $\tau$ adapted to $X_1,\ldots ,X_n$ that maximizes $P(X_{\tau }=M_n)$? This problem was examined by Gilbert and Mosteller (J Am Stat Assoc 61:35–73, 1966) and the optimal win probability in this case is denoted by $v_{n,\mathrm{max}}^*$. What if it is desired to “stop within a proportion of the maximum”? That is, for $0<\alpha <1$, what is the stopping rule $\tau$ that maximizes $P(X_{\tau } \ge \alpha M_n)$? In this note it is proven that for any continuous random variable X if $\tau ^*$ is the optimal stopping rule then $P(X_{\tau ^*} \ge \alpha M_n)\ge v_{n,\mathrm{max}}^*$, and this lower bound is sharp. Some examples and another interesting result are presented.