@article{a6c592e27d6f4b1aa9c05bc8ecbae7dc,
title = "Halfspace depths for scatter, concentration and shape matrices",
abstract = "We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept is similar to those from Chen, Gao and Ren [Robust covariance and scatter matrix estimation under Huber{\textquoteright}s contamination model (2018)] and Zhang [J. Multivariate Anal. 82 (2002) 134–165]. Rather than focusing, as in these earlier works, on deepest scatter matrices, we thoroughly investigate the properties of the proposed depth and of the corresponding depth regions. We do so under minimal assumptions and, in particular, we do not restrict to elliptical distributions nor to absolutely continuous distributions. Interestingly, fully understanding scatter halfspace depth requires considering different geometries/topologies on the space of scatter matrices. We also discuss, in the spirit of Zuo and Serfling [Ann. Statist. 28 (2000) 461–482], the structural properties a scatter depth should satisfy, and investigate whether or not these are met by scatter halfspace depth. Companion concepts of depth for concentration matrices and shape matrices are also proposed and studied. We show the practical relevance of the depth concepts considered in a real-data example from finance.",
keywords = "Curved parameter spaces, Elliptical distributions, Robustness, Scatter matrices, Shape matrices, Statistical depth",
author = "Davy Paindaveine and {Van Bever}, Germain",
note = "Funding Information: Supported by the IAP research network Grant Nr. P7/06 of the Belgian government (Belgian Science Policy), the Cr{\'e}dit de Recherche J.0113.16 of the FNRS (Fonds National pour la Recherche Scientifique), Communaut{\'e} Fran{\c c}aise de Belgique and a grant from the National Bank of Belgium. 2Supported by the FC84444 Grant of the FNRS (Fonds National pour la Recherche Scientifique), Communaut{\'e} Fran{\c c}aise de Belgique. Funding Information: Received April 2017; revised October 2017. 1Supported by the IAP research network Grant Nr. P7/06 of the Belgian government (Belgian Science Policy), the Cr{\'e}dit de Recherche J.0113.16 of the FNRS (Fonds National pour la Recherche Scientifique), Communaut{\'e} Fran{\c c}aise de Belgique and a grant from the National Bank of Belgium. 2Supported by the FC84444 Grant of the FNRS (Fonds National pour la Recherche Scientifique), Communaut{\'e} Fran{\c c}aise de Belgique. MSC2010 subject classifications. Primary 62H20; secondary 62G35. Key words and phrases. Curved parameter spaces, elliptical distributions, robustness, scatter matrices, shape matrices, statistical depth. Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2018. Copyright: Copyright 2018 Elsevier B.V., All rights reserved.",
year = "2018",
doi = "10.1214/17-AOS1658",
language = "English",
volume = "46",
pages = "3276--3307",
journal = "Annals of Statistics",
issn = "0090-5364",
publisher = "Institute of Mathematical Statistics",
number = "6B",
}