ABSTRACT

The unit interval $[0,1]$ is broken at random into n spacings with the breaking points given by a random sample of size n−1 from the uniform distribution on $[0,1]$. A decision-maker observes the lengths of these spacings sequentially and must decide either to select the present spacing or to reject it and continue to observe the next one. No recall of preceding observations is permitted. In this paper, we first find an optimal stopping rule to maximize the probability of selecting the largest spacing. Furthermore, we establish a connection to the classical secretary problem, from which we derive a lower bound for the maximum probability. Moreover, we conjecture that the limiting optimal probability is the same as in the full-information best-choice problem. Next, we find an optimal stopping rule to maximize the expected length of the selected spacing. It is shown that the maximum obtainable expected length lies between the expected value of the second-largest and the largest spacing.

抽象的

单位间隔 $[0，\mathrm{1个}]$被随机分成n个间距，其断裂点由n -1的均匀分布从n -1的随机样本给出$[0，\mathrm{1个}]$。决策者顺序观察这些间距的长度，必须决定选择当前间距还是拒绝当前间距，并继续观察下一个间距。不允许召回先前的观察结果。在本文中，我们首先找到一个最佳的停止规则，以使选择最大间距的概率最大化。此外，我们建立了与古典秘书问题的联系，从中我们得出最大概率的下界。此外，我们推测限制的最佳概率与完整信息的最佳选择问题相同。接下来，我们找到了一个最佳的停止规则，以使所选间距的预期长度最大化。结果表明，最大可获得的期望长度在第二最大间距与最大间距的期望值之间。