SCoRE Sets: A Versatile Framework for Simultaneous Inference

TL;DR

This paper introduces SCoRE Sets, a flexible framework for simultaneous inference on functions over metric spaces, unifying various statistical tools like confidence bands and multiple testing under a common approach.

Contribution

It develops the concept of SCoRE Sets for asymptotic inference, connecting and refining existing methods such as CoPE sets and relevance testing with fewer assumptions.

Findings

SCoRE Sets unify multiple inference tools under a single framework.

They provide refined, assumption-light methods for simultaneous confidence and hypothesis testing.

The framework enhances the understanding and application of level set and relevance testing.

Abstract

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.