Extremal Depth for Functional Data and Applications

ABSTRACT

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.

KEYWORDS:

• Central regions
• Data depth
• Functional boxplots
• Outlier detection
• Simultaneous inference

Supplementary Materials

The supplementary material contains a more general definition of extremal depth followed by conditions and proofs for all the theoretical results in the article.

Acknowledgments

The authors are grateful to the editors and the referees for their comments and suggestions that have led to significant improvements of an earlier version of this article.