TY - JOUR
T1 - From depth to local depth
T2 - A focus on centrality
AU - Paindaveine, Davy
AU - Van Bever, Germain
PY - 2013/12/16
Y1 - 2013/12/16
N2 - 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.
AB - 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.
KW - Classification
KW - Multimodality
KW - Nonconvex support
KW - Regression depth
KW - Statistical depth functions
KW - Symmetry testing
UR - http://www.scopus.com/inward/record.url?scp=84890111025&partnerID=8YFLogxK
U2 - 10.1080/01621459.2013.813390
DO - 10.1080/01621459.2013.813390
M3 - Article
AN - SCOPUS:84890111025
SN - 0162-1459
VL - 108
SP - 1105
EP - 1119
JO - Journal of the American Statistical Association
JF - Journal of the American Statistical Association
IS - 503
ER -