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——一种基础探测模型，其目标是从$n$个独立同分布随机变量中选出较大值。算法可为每个变量$X_i$设置阈值$t_i$进行一次测试，返回该变量是否满足$X_i \ge t_i$的二元反馈。阈值可由算法自适应或非自适应地选择。根据每个变量的测试结果，我们选择条件期望最高的变量。我们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。事实上，当$n \to \infty$时，非自适应算法在阈值测试中的最优性能由标准独立同分布先知不等式（约$0.745+o(1)$）决定。我们展示了自适应算法如何显著提升该比率：我们的自适应测试策略可保证至少$0.869-o(1)$的竞争比。此外，我们证明即使当$n \to \infty$时，某些分布仅允许常数比$c < 1$。最后，当每个箱可被多次测试（总计$n$次测试）时，我们设计了一个可实现$1-o(1)$比率的算法。