Title: Simultaneous confidence bands and testing for functional data using the GKF with Applications to DTI fibers
Authors: Fabian Telschow - University of California San Diego (United States) [presenting]
Armin Schwartzman - University of California, San Diego (United States)
Abstract: The use of the Gaussian Kinematic formula (GKF) is explored for constructing simultaneous confidence bands (SCBs) of the population mean curve in densely observed functional data. The GKF is a non-asymptotic formula -- hence valid also for small sample sizes --, which is exploited to estimate thresholds of the maximum of a pointwise $t$-statistic yielding SCBs. Although the GKF relies on smooth Gaussian processes, we show that -- having a CLT for the estimator of the mean curve -- our approach produces SCBs with asymptotically precise covering rates even under non-Gaussian errors and observation noise. Moreover, in order to avoid bandwidth choices in practice, we borrow ideas from scale spaces, and generalize the above methodology to obtain SCBs of the smoothed population means valid simultaneously across many bandwidths. The performance of this method is compared to state of the art procedures with a focus on small sample sizes ($N\approx10-30$), where under the Gaussian paradigm our method gives precise covering rates and outperforms its competitors including bootstrap methods. Additionally, we provide an application to a DTI fiber data set of $15$ healthy controls and $15$ patients. The method is applied to find and localize significant differences between the mean curves of the two groups.