Fast and Fair Simultaneous Confidence Bands for Functional Parameters

TL;DR

This paper introduces a fast, fair, and covariance-free method for constructing simultaneous confidence bands for functional parameters, enabling better uncertainty quantification in functional data analysis with practical applications.

Contribution

The paper presents a novel methodology for confidence bands that are computationally efficient, fair across the domain, and applicable to fragmentary data, filling gaps in existing approaches.

Findings

Bands are computationally fast due to no resampling.

They maintain balanced false positive rates across the domain.

Simulations show superior finite-sample performance.

Abstract

Quantifying uncertainty using confidence regions is a central goal of statistical inference. Despite this, methodologies for confidence bands in Functional Data Analysis are still underdeveloped compared to estimation and hypothesis testing. In this work, we present a new methodology for constructing simultaneous confidence bands for functional parameter estimates. Our bands possess a number of positive qualities: (1) they are not based on resampling and thus are fast to compute, (2) they are constructed under the fairness constraint of balanced false positive rates across partitions of the bands' domain which facilitates the typical global, but also novel local interpretations, and (3) they do not require an estimate of the full covariance function and thus can be used in the case of fragmentary functional data. Simulations show the excellent finite-sample behavior of our bands in comparison to existing alternatives. The practical use of our bands is demonstrated in two case studies on sports biomechanics and fragmentary growth curves.