We study the problem of finding the index of the minimum value of a vector from noisy observations. This problem is relevant in population/policy comparison, discrete maximum likelihood, and model selection. We develop an asymptotically normal test statistic, even in high-dimensional settings and with potentially many ties in the population mean vector, by integrating concepts and tools from cross-validation and differential privacy. The key technical ingredient is a central limit theorem for globally dependent data. We also propose practical ways to select the tuning parameter that adapts to the signal landscape. Numerical experiments and data examples demonstrate the ability of the proposed method to achieve a favorable bias-variance trade-off in practical scenarios.

翻译：我们研究从噪声观测中寻找向量最小值索引的问题。该问题在群体/策略比较、离散极大似然估计和模型选择中具有重要意义。通过整合交叉验证和差分隐私的概念与工具，我们构建了一个渐近正态的检验统计量，即使在高维设置下且总体均值向量可能存在大量相等值的情况下依然适用。关键技术要素是针对全局依赖数据的中心极限定理。我们还提出了适应信号分布特性的调参参数实用选择方法。数值实验和实际数据案例表明，所提方法能够在实际场景中实现良好的偏差-方差权衡。