损失与秩相关的最优停止
Optimal Stopping with Rank-Dependent Loss
摘要：
对于$τ$，一个适合于$n$ iid观测序列的停止规则，我们将损失定义为$\ex [ q(R_τ)]$，其中$R_j$是第$j$个观测的秩，$q$是秩的非减函数。此设置涵盖了使用$q(r)={\bf 1}(r>1)$的最佳选择问题和使用$q(r)=r$的罗宾斯问题。作为$n\to\infty$，停止问题获得了与平面泊松过程相关联的极限形式。通过对极限的考察，我们建立了停止值的界，并揭示了最优规则的定性特征。特别是，我们表明，完整的历史依赖性持续的限制，从而回答了一个问题，问的背景下罗宾斯的问题。
英文摘要：
For $τ$ a stopping rule adapted to a sequence of $n$ iid observations, we define the loss to be $\ex [ q(R_τ)]$, where $R_j$ is the rank of the $j$th observation, and $q$ is a nondecreasing function of the rank. This setting covers both the best choice problem with $q(r)={\bf 1}(r>1)$, and Robbins' problem with $q(r)=r$. As $n\to\infty$ the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit, thus answering a question asked by Bruss in the context of Robbins' problem.
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