The Annals of Statistics

Halfspace depths for scatter, concentration and shape matrices

Davy Paindaveine and Germain Van Bever

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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.

Article information

Ann. Statist., Volume 46, Number 6B (2018), 3276-3307.

Received: April 2017
Revised: October 2017
First available in Project Euclid: 11 September 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Secondary: 62G35: Robustness

Curved parameter spaces elliptical distributions robustness scatter matrices shape matrices statistical depth


Paindaveine, Davy; Van Bever, Germain. Halfspace depths for scatter, concentration and shape matrices. Ann. Statist. 46 (2018), no. 6B, 3276--3307. doi:10.1214/17-AOS1658.

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Supplemental materials

  • Supplement to “Halfspace depths for scatter, concentration and shape matrices”. In this supplement, we conduct a Monte Carlo exercise validating the explicit scatter halfspace depth expressions obtained in the Gaussian and independent Cauchy examples. We also provide illustrations of Theorem 3.3 and Theorems 7.8–7.10. Finally, we prove all theorems stated in this paper.