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arXiv math/0510242math.PRmath.STstat.TH
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二选最优停止

Two choice optimal stopping

David Assaf, Larry Goldstein, Ester Samuel-Cahn

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33 pages
发表信息：
Advances in Applied Probability 2004, Vol 36, 1116-1147
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摘要：

设$X_n,...,X_1$为i.i.d.。分布函数为$F$的随机变量。统计学家知道$F$，按顺序观察$X$值，并有两次机会使用停止规则选择$X$。统计学家的目标是在尽可能小的值$X$处停止。令$V_n^2$等于统计学家在进行优化时选择的两个值中较小者的期望值。我们得到了序列$V_n^2$的渐近行为的一个大类的$F$的属于吸引区域（最小值）${\cal D}(G^α)$，其中$G^α(x)=[1-\exp(-x^α)]{\bf I}(x \ge 0)$。将所得结果与经典的单选值序列$V_n^1$的渐近行为以及与"先知值”序列$V_n^p=E(\min\{X_n,...,X_1\})$的渐近行为进行了比较.

英文摘要：

Let $X_n,...,X_1$ be i.i.d. random variables with distribution function $F$. 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 stop at a value of $X$ as small as possible. Let $V_n^2$ equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behavior of the sequence $V_n^2$ for a large class of $F$'s belonging to the domain of attraction (for the minimum) ${\cal D}(G^α)$, where $G^α(x)=[1-\exp(-x^α)]{\bf I}(x \ge 0)$. The results are compared with those for the asymptotic behavior of the classical one choice value sequence $V_n^1$, as well as with the ``prophet value" sequence $V_n^p=E(\min\{X_n,...,X_1\})$.

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