We study ranking and selection under input uncertainty in settings where additional data cannot be collected. We propose the Nonparametric Input-Output Uncertainty Comparisons (NIOU-C) procedure to construct a confidence set that includes the optimal solution with a user-specified probability. We construct an ambiguity set of input distributions using empirical likelihood and approximate the mean performance of each solution using a linear functional representation of the input distributions. By solving optimization problems evaluating worst-case pairwise mean differences within the ambiguity set, we build a confidence set of solutions indistinguishable from the optimum. We characterize sample size requirements for NIOU-C to achieve the asymptotic validity under mild conditions. Moreover, we propose an extension to NIOU-C, NIOU-C:E, that mitigates conservatism and yields a smaller confidence set. In numerical experiments, NIOU-C provides a smaller confidence set that includes the optimum more frequently than a parametric procedure that takes advantage of the parametric distribution families.

翻译：本研究探讨在无法收集额外数据的情况下，输入不确定性下的排序与选择问题。我们提出非参数输入-输出不确定性比较（NIOU-C）方法，以构建一个在用户指定概率下包含最优解的置信集。通过经验似然构建输入分布的模糊集，并利用输入分布的线性泛函表示来近似各解决方案的平均性能。通过求解在模糊集内评估最坏情况成对均值差异的优化问题，我们构建了一个与最优解不可区分的解决方案置信集。我们刻画了NIOU-C在温和条件下达到渐近有效性所需的样本量要求。此外，我们提出NIOU-C的扩展版本NIOU-C:E，以减轻保守性并获得更小的置信集。数值实验表明，与利用参数分布族的参数化方法相比，NIOU-C能提供更小的置信集，且更频繁地包含最优解。