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February 2016 A Topologically Valid Definition of Depth for Functional Data
Alicia Nieto-Reyes, Heather Battey
Statist. Sci. 31(1): 61-79 (February 2016). DOI: 10.1214/15-STS532

Abstract

The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.

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Alicia Nieto-Reyes. Heather Battey. "A Topologically Valid Definition of Depth for Functional Data." Statist. Sci. 31 (1) 61 - 79, February 2016. https://doi.org/10.1214/15-STS532

Information

Published: February 2016
First available in Project Euclid: 10 February 2016

zbMATH: 06946212
MathSciNet: MR3458593
Digital Object Identifier: 10.1214/15-STS532

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.31 • No. 1 • February 2016
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