Inverse set estimation and inversion of simultaneous confidence intervals

TL;DR

This paper introduces a method for estimating sets in a function's domain based on confidence intervals, applicable in risk assessment contexts like climatology and medicine, with guarantees against data peeking.

Contribution

It generalizes set estimation to dense and non-dense domains and provides a non-parametric bootstrap algorithm with simultaneous confidence levels.

Findings

Confidence sets can be constructed non-asymptotically for multiple levels.

Method applies to dense and non-dense domains.

Provides a bootstrap algorithm and code implementation.

Abstract

Motivated by the questions of risk assessment in climatology (temperature change in North America) and medicine (impact of statin usage and COVID-19 on hospitalized patients), we address the problem of estimating the set in the domain of a function whose image equals a predefined subset. Existing methods that construct confidence sets require strict assumptions. We generalize the estimation of such sets to dense and non-dense domains with protection against "data peeking" by proving that confidence sets of multiple levels can be simultaneously constructed with the desired confidence non-asymptotically through inverting simultaneous confidence bands. A non-parametric bootstrap algorithm and code are provided.