  We propose a new notion called `extremal depth' (ED) for functional data,
discuss its properties, and compare its performance with existing concepts. The
proposed notion is based on a measure of extreme `outlyingness'. ED has several
desirable properties that are not shared by other notions and is especially
well suited for obtaining central regions of functional data and function
spaces. In particular: a) the central region achieves the nominal (desired)
simultaneous coverage probability; b) there is a correspondence between
ED-based (simultaneous) central regions and appropriate point-wise central
regions; and c) the method is resistant to certain classes of functional
outliers. The paper examines the performance of ED and compares it with other
depth notions. Its usefulness is demonstrated through applications to
constructing central regions, functional boxplots, outlier detection, and
simultaneous confidence bands in regression problems.
