Simultaneous hypothesis testing for comparing many functional means

TL;DR

This paper introduces a new statistical test for comparing many functional means simultaneously, controlling error rates without dimension reduction, and is applicable to high-dimensional functional data like fMRI.

Contribution

It develops a fully functional paired two-sample test that handles increasing numbers of functional recordings and provides theoretical guarantees through new Gaussian approximation results.

Findings

Test effectively controls family-wise error rate.

Applicable to high-dimensional functional data like fMRI.

Theoretical results on maxima of $L^2$ statistics.

Abstract

Data with multiple functional recordings at each observational unit are increasingly common in various fields including medical imaging and environmental sciences. To conduct inference for such observations, we develop a paired two-sample test that allows to simultaneously compare the means of many functional observations while maintaining family-wise error rate control. We explicitly allow the number of functional recordings to increase, potentially much faster than the sample size. Our test is fully functional and does not rely on dimension reduction or functional PCA type approaches or the choice of tuning parameters. To provide a theoretical justification for the proposed procedure, we develop a number of new anti-concentration and Gaussian approximation results for maxima of $L^2$ statistics which might be of independent interest. The methodology is illustrated on the task-related cortical surface functional magnetic resonance imaging data from Human Connectome Project.