Simultaneous confidence bands for the mean of functional data

The mean function is a central object of inquiry in the analysis of functional data. Typical questions related to the mean function include quantifying estimation uncertainty, testing parametric models, and making comparisons between populations. To make probabilistic statements about the mean function over its entire domain, rather than at a single location, it is necessary to infer all of its values simultaneously. Pointwise inference is not appropriate for this task and indeed produces anticonservative results, i.e., the coverage level of confidence regions is too low and the significance level of hypothesis tests too high. In contrast, simultaneous confidence bands (SCB) provide a flexible framework for conducting simultaneous inference on the mean function and other functional parameters. They also offer powerful visualization tools for communicating analytic results to interdisciplinary audiences. The construction of SCB in the context of functional data requires specific theory and methods. In particular, it is not addressed by the nonparametric regression literature. Although software is available to perform individual steps of an SCB procedure, resources that provide end-to-end computations are scarce. Applications of SCB to one- and two-sample inferences are illustrated here with the R package SCBmeanfd. WIREs Comput Stat 2017, 9:e1397. doi: 10.1002/wics.1397. For further resources related to this article, please visit the WIREs website.