@article{10902/9546,
year = {2016},
month = {2},
url = {http://hdl.handle.net/10902/9546},
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.},
organization = {Alicia Nieto-Reyes is grateful to the School of Mathematics
at the University of Bristol for kind hospitality
during the 2013/2014 academic year, during which
time this work was carried out. We thank Juan Cuesta-
Albertos and Peter Green for their comments on a
single-authored unpublished manuscript, from which
the seeds of this paper originated, and Peter Hall for
kindly reading the final draft. We also thank three
anonymous referees for constructive suggestions.
Alicia Nieto-Reyes supported in part by the Spanish
Ministerio de Ciencia e Innovación under grant
MTM201128657-C0202. Heather Battey supported in part by the EPSRC under grant EP/D063485/1.},
publisher = {Institute of Mathematical Statistics (IMS)},
publisher = {Statistical Science, 2016, 31(1), 61-79 - (CORRIGENDUM), 2017, 32(4), 640.},
title = {A topologically valid definition of depth for functional data},
author = {Nieto Reyes, Alicia and Battey, Heather},
}