We study asymptotic statistical inference in the space of bounded functions endowed with the supremum norm over an arbitrary metric space $S$ using a novel concept: Simultaneous Confidence Probability Excursion (SCoPE) sets. Given an estimator SCoPE sets simultaneously quantify the uncertainty of several lower and upper excursion sets of a target function and thereby 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 $S$. As a byproduct our abstract treatment allows us to refine and generalize the methodology and reduce the assumptions in recent articles in relevance and equivalence testing in functional data.

翻译：我们研究在带有一致范数的有界函数空间中，于任意度量空间$S$上的渐近统计推断，提出一种新概念：同时置信概率波动（SCoPE）集。给定一个估计量，SCoPE集能同时量化目标函数多个下水平和上水平波动集的不确定性，从而为多种统计推断工具提供统一视角，例如同时置信带、水平集估计中的不确定性量化（如CoPE集），以及$S$上的多重假设检验（如寻找$S$内的相关差异或等价区域）。作为副产品，我们的抽象处理方法使我们能够改进和推广近期关于函数数据相关性与等价性检验的多篇论文中的方法，并减少其假设条件。