# Simultaneous Confidence Bands for Functional Data Using the Gaussian Kinematic Formula

## TL;DR

This paper develops a method for constructing simultaneous confidence bands for functional data using the Gaussian Kinematic formula, demonstrating its effectiveness even with small samples and extending it to noisy, discretely sampled data.

## Contribution

It introduces a CLT-based approach for applying the Gaussian Kinematic formula to non-Gaussian functional data, with practical extensions and a comprehensive simulation study.

## Key findings

- tGKF performs well with small sample sizes
- tGKF outperforms bootstrap methods for larger samples
- Method is computationally efficient and applicable to DTI data

## Abstract

This article constructs simultaneous confidence bands (SCBs) for functional parameters using the Gaussian Kinematic formula of $t$-processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering rates even for non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipschitz-Killing curvatures, the only data dependent quantities in the tGKF SCBs. Extensions to discrete sampling with additive observation noise are discussed using scale space ideas from regression analysis. Here we provide sufficient conditions on the processes and kernels to obtain convergence of the functional scale space surface. The theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF works well even for small sample sizes and only a Rademacher multiplier-$t$ bootstrap performs similarily well. For larger sample sizes the tGKF often outperforms the bootstrap methods and is computational faster. We apply the method to diffusion tensor imaging (DTI) fibers using a scale space approach for the difference of population means. R code is available in our Rpackage SCBfda.