Fast and Fair Simultaneous Confidence Bands for Functional Parameters

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 striking qualities: (1) they have a nearly closed-form expression and thus are fast to compute, (2) they can be constructed adaptively according to a desired criterion, where we focus on the fairness constraint of false positive rate balance across partitions of the bands' domain which facilitates both global and 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.