Extremal Depth for Functional Data and Applications
提出了一种用于函数型数据的“极值深度”概念，基于极端离群度测量，能构建同时覆盖概率名义水平的中心区域，并应用于中心区域构建、函数箱线图、异常检测和回归的同时置信带。
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 pointwise central regions; and (c) the method is resistant to certain classes of functional outliers. The article 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. Supplementary materials for this article are available online.