Open Access
December 2018 Halfspace depths for scatter, concentration and shape matrices
Davy Paindaveine, Germain Van Bever
Ann. Statist. 46(6B): 3276-3307 (December 2018). DOI: 10.1214/17-AOS1658


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’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.


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Davy Paindaveine. Germain Van Bever. "Halfspace depths for scatter, concentration and shape matrices." Ann. Statist. 46 (6B) 3276 - 3307, December 2018.


Received: 1 April 2017; Revised: 1 October 2017; Published: December 2018
First available in Project Euclid: 11 September 2018

zbMATH: 1408.62100
MathSciNet: MR3852652
Digital Object Identifier: 10.1214/17-AOS1658

Primary: 62H20
Secondary: 62G35

Keywords: Curved parameter spaces , elliptical distributions , robustness , scatter matrices , shape matrices , Statistical depth

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 6B • December 2018
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