Department of Statistics
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Title: An Extremal Notion of Functional Depth
Notions of data depth for multivariate data have been studied extensively and their usefulness has been demonstrated in a variety of applications. However, depth notions for infinite dimensional objects, such as functional data, have received relatively less attention. We propose a new notion called extremal depth for functional data. In contrast to existing notions, such as integrated data depth and band depth, the proposed notion considers extreme ?¡ãoutlyingness?¡À, similar to the projection depth in the multivariate case. Extremal depth satisfies many of the natural properties for depth and is well suited to the purpose of obtaining central regions of functional data and distributions on functional spaces. Moreover, when used for constructing central regions of a functional distribution of interest, it achieves desired coverage, unlike other existing notions which have over-coverage. Another novel feature is that the central regions obtained have width that is proportional to a measure of variability of the random functions at different points in the domain. A number of simulated and real examples are used to demonstrate the usefulness of the method. This is joint work Prof. Vijay Nair at the University of Michigan.