From Depth to Local Depth: A Focus on Centrality

Abstract

Aiming at analyzing multimodal or nonconvexly supported distributions through data depth, we introduce a local extension of depth. Our construction is obtained by conditioning the distribution to appropriate depth-based neighborhoods and has the advantages, among others, of maintaining affine-invariance and applying to all depths in a generic way. Most importantly, unlike their competitors, which (for extreme localization) rather measure probability mass, the resulting local depths focus on centrality and remain of a genuine depth nature at any locality level. We derive their main properties, establish consistency of their sample versions, and study their behavior under extreme localization. We present two applications of the proposed local depth (for classification and for symmetry testing), and we extend our construction to the regression depth context. Throughout, we illustrate the results on several datasets, both artificial and real, univariate and multivariate. Supplementary materials for this article are available online.

KEY WORDS:

• Classification
• Multimodality
• Nonconvex support
• Regression depth
• Statistical depth functions
• Symmetry testing

Acknowledgments

He is also a member of ECORE, the association between CORE and ECARES. Davy Paindaveine's research is supported by an Action de Recherche Concertée (A.R.C.) contract from the Communauté Française de Belgique and by the InterUniversity Attraction Pole (IAP) research network grant nr. P7/06 of the Belgian government (Belgian Science Policy). Germain Van Bever's research is supported through a Mandat d’Aspirant FNRS (Fonds National pour la Recherche Scientifique), Communauté Française de Belgique. The authors are grateful to three anonymous referees, an Associate Editor, and the Editor Xuming He for their careful reading and insightful comments that led to substantial improvements of the article. They also wish to thank Claudio Agostinelli for stimulating discussions.