Confidence Bands in Functional data - The Bootstrap or Gaussian Kinematic formula?

In this talk we study simultaneous confidence bands (SCBs) for functional parameters. We introduce a new Multiplier Bootstrap and a “parametric approach” using the Gaussian Kinematic Formula (GKF) for construction of SCBs. The GKF as introduced by Jonathan Taylor can be use to approximate the distribution of the maximum of Gaussian related processes for large thresholds. One of the main results of this talk will be an error bound on the asymptotical coverage rate of SCBs constructed using the GKF, which basically requires only a functional CLT for the estimator of the functional parameter and some regularity assumptions on the limiting process.

We also shortly discuss a strategy how these ideas can be extended to discretely observed functional processes contaminated by observation noise, where we build on Scale Spaces introduced by Chaudhuri and Marron in the early 2000’s.

The theoretical discussion will be accompanied by simulation studies for the population mean in signal plus noise models and an application of a two sample situation in DTI fibers. In the end we will give a short outlook on different settings our method can also be applied to, e.g. Signal-to-Noise ratios (Cohen’s d) or General linear Models, which are of interest in statistical analysis of fMRI data.