Extremal Depth for Functional Data and Applications

TL;DR

This paper introduces extremal depth (ED), a new measure for functional data that effectively identifies central regions, detects outliers, and constructs confidence bands, outperforming existing methods in robustness and coverage accuracy.

Contribution

The paper presents extremal depth (ED), a novel depth measure for functional data with desirable properties like robustness and accurate coverage, along with applications to various statistical tools.

Findings

ED provides accurate simultaneous coverage probabilities.

ED-based central regions correspond well with point-wise regions.

The method is resistant to certain functional outliers.

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 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.