Small subsets of data with disproportionate influence on model outcomes can have dramatic impacts on conclusions, with a few data points sometimes overturning key findings. While recent work has developed methods to identify these most influential sets, no formal theory exists to determine when their influence reflects genuine problems rather than natural sampling variation. We address this gap by developing a principled framework for assessing the statistical significance of most influential sets. Our theoretical results characterize the extreme value distributions of maximal influence and enable rigorous hypothesis tests for excessive influence, replacing current ad-hoc sensitivity checks. We demonstrate the practical value of our approach through applications across economics, biology, and machine learning benchmarks.

翻译：数据中少数具有不成比例影响力的子集可能对模型结果产生显著影响，有时仅需几个数据点即可推翻关键结论。尽管近期研究已开发出识别这些最具影响力集合的方法，但目前尚缺乏正式理论来判断其影响力何时反映真实问题而非自然抽样变异。我们通过建立评估最具影响力集合统计显著性的理论框架来填补这一空白。我们的理论结果刻画了最大影响力的极值分布特征，并为过度影响力提供了严格的假设检验方法，从而取代当前临时性的敏感性检查。通过经济学、生物学及机器学习基准测试等领域的应用案例，我们验证了该方法的实用价值。