Simultaneous confidence bands for functional data using the Gaussian Kinematic formula

Abstract

We propose a construction of simultaneous confidence bands (SCBs) for functional parameters over arbitrary dimensional compact domains 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 even if the observations are 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 Lipshitz–Killing curvatures, the only data-dependent quantities in the tGKF. We further discuss extensions to discrete sampling with additive observation noise using scale space ideas from regression analysis. Our theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF outperforms state-of-the-art methods with precise covering for small sample sizes, and only a Rademacher multiplier-t bootstrap performs similarly well. A further benefit is that our SCBs are computational fast even for domains of dimension greater than one. Applications of SCBs to diffusion tensor imaging (DTI) fibers (1D) and spatio-temporal temperature data (2D) are discussed.