We study threshold testing, an elementary probing model with the goal to choose a large value out of $n$ i.i.d. random variables. An algorithm can test each variable $X_i$ once for some threshold $t_i$, and the test returns binary feedback whether $X_i \ge t_i$ or not. Thresholds can be chosen adaptively or non-adaptively by the algorithm. Given the results for the tests of each variable, we then select the variable with highest conditional expectation. We compare the expected value obtained by the testing algorithm with expected maximum of the variables. Threshold testing is a semi-online variant of the gambler's problem and prophet inequalities. Indeed, the optimal performance of non-adaptive algorithms for threshold testing is governed by the standard i.i.d. prophet inequality of approximately $0.745+o(1)$ as $n \to \infty$. We show how adaptive algorithms can significantly improve upon this ratio. Our adaptive testing strategy guarantees a competitive ratio of at least $0.869-o(1)$. Moreover, we show that there are distributions that admit only a constant ratio $c < 1$, even when $n \to \infty$. Finally, when each box can be tested multiple times (with $n$ tests in total), we design an algorithm that achieves a ratio of $1-o(1)$.

翻译：我们研究阈值测试——一种旨在从$n$个独立同分布随机变量中选取较大值的基本探测模型。算法可对每个变量$X_i$以特定阈值$t_i$进行一次测试，测试返回二值反馈：$X_i \ge t_i$或反之。算法可自适应或非自适应地选择阈值。根据各变量测试结果，我们选择条件期望最高的变量。将测试算法获得的期望值与变量的期望最大值进行比较。阈值测试是赌徒问题与先知不等式的半在线变体。实际上，随着$n \to \infty$，非自适应算法在阈值测试中的最优性能受限于标准的独立同分布先知不等式，该比值约为$0.745+o(1)$。我们展示自适应算法如何显著提升此比值：所提出的自适应测试策略保证竞争比至少达到$0.869-o(1)$。此外，我们证明存在某些分布，即使当$n \to \infty$时，其竞争比也仅为常数$c < 1$。最后，当每个箱子可进行多次测试（总计$n$次测试）时，我们设计了一种实现$1-o(1)$比值的算法。