Why do these quite different best-choice problems have the same solutions?
Why do these quite different best-choice problems have the same solutions?
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为什么这些截然不同的最佳选择问题有相同的解决方案？
DOI：
10.1239/aap/1086957578
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发表时间：
2004
影响因子：
1.2
通讯作者：
S. M. Samuels
中科院分区：
文献类型：
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作者：
S. M. Samuels
吉尔伯特和莫斯特勒在1966年提出的全信息最佳选择问题要求我们找到一个停止规则，使选择n个独立同分布序列中最大者的概率最大化。标准均匀随机变量1987年，Porosiberski用一个随机N代替了一个固定的n，在{1，2，.，n}上是均匀的，并且与观测值无关。一个部分信息的问题，嵌入在1980年的文件Petruccelli，保持n固定，但允许我们观察的范围（最大-最小），以及是否目前的观察是最大的。最近，Porosibowski比较了他和Petruccelli问题的解决方案，发现这两个问题具有相同的最优规则以及渐近相等的风险。他的发现提出了一个问题：为什么？本文很好地解释了最优规则的等价性。但是，即使在平面泊松过程模型的透镜下，它也使渐近风险的等效性成为一个谜。与此同时，另外两个问题也被证明具有相同的极限风险：（次优的）Porosibiski-Petruccelli停止规则的全信息问题，以及Ferguson，Hardwick和Tamaki的全信息“最佳持续时间”问题，这只是Porosibiski-Petruccelli问题的伪装。
The full-information best-choice problem, as posed by Gilbert and Mosteller in 1966, asks us to find a stopping rule which maximizes the probability of selecting the largest of a sequence of n i.i.d. standard uniform random variables. Porosiński, in 1987, replaced a fixed n by a random N, uniform on {1,2,…,n} and independent of the observations. A partial-information problem, imbedded in a 1980 paper of Petruccelli, keeps n fixed but allows us to observe only the sequence of ranges (max - min), as well as whether or not the current observation is largest so far. Recently, Porosiński compared the solutions to his and Petruccelli's problems and found that the two problems have identical optimal rules as well as risks that are asymptotically equal. His discovery prompts the question: why? This paper gives a good explanation of the equivalence of the optimal rules. But even under the lens of a planar Poisson process model, it leaves the equivalence of the asymptotic risks as somewhat of a mystery. Meanwhile, two other problems have been shown to have the same limiting risks: the full-information problem with the (suboptimal) Porosiński-Petruccelli stopping rule, and the full-information ‘duration of holding the best’ problem of Ferguson, Hardwick and Tamaki, which turns out to be nothing but the Porosiński problem in disguise.