We study asymptotic statistical inference in the space of bounded functions endowed with the supremums norm over an arbitrary metric space $S$ using a novel concept: Simultaneous COnfidence Region of Excursion (SCoRE) Sets. They simultaneously quantify the uncertainty of several lower and upper excursion sets of a target function. We investigate their connection to multiple hypothesis tests controlling the familywise error rate in the strong sense and show that they grant a unifying perspective on several statistical inference tools such as simultaneous confidence bands, quantification of uncertainties in level set estimation, for example, CoPE sets, and multiple hypothesis testing over $S$, for example, finding relevant differences or regions of equivalence within $S$. In particular, our abstract setting allows us to refine and reduce the assumptions in recent articles on CoPE sets and relevance and equivalence testing using the supremums norm.

翻译：我们利用一种新概念——同步置信区域集合（SCoRE集合），在任意度量空间$S$上，以有界函数空间配备上确界范数研究渐近统计推断。该集合能够同时量化目标函数多个下、上偏移集合的不确定性。我们探讨了SCoRE集合与强意义下控制族错误率的多元假设检验之间的关联，并证明其为多种统计推断工具提供了统一视角，例如同步置信带、水平集估计中的不确定性量化（如CoPE集合）以及$S$上的多元假设检验（例如寻找$S$内的相关差异或等价区域）。特别地，我们的抽象框架使得能够在使用上确界范数的CoPE集合及相关性与等价性检验的最新研究中，优化并减少其假设条件。