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：我们利用一种新颖概念——同步置信概率波动（SCoPE）集，研究在任意度量空间$S$上赋予上确界范数的有界函数空间中的渐近统计推断。给定估计量后，SCoPE集可同步量化目标函数多个低层与高层波动集的不确定性，从而为多种统计推断工具提供统一视角，包括同步置信带、水平集估计中的不确定性量化（例如CoPE集），以及$S$上的多重假设检验（例如寻找$S$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能够优化并推广相关方法论，并减少近期关于函数型数据相关性与等价性检验文献中的假设条件。