From Depth to Local Depth: A Focus on Centrality

Aiming at analysing multimodal or non-convexly supported distributionsthrough data depth, we introduce a local extension of depth. Our constructionis obtained by conditioning the distribution to appropriate depth-based neigh-borhoods, and has the advantages, among others, to maintain a_ne-invarianceand to apply to all depths in a generic way. Most importantly, unlike their com-petitors, that (for extreme localization) rather measure probability mass, theresulting local depths focus on centrality and remain of a genuine depth natureat any locality level. We derive their main properties, establish consistency oftheir sample versions, and study their behavior under extreme localization. Wealso extend our local concept to the regression and functional depth contexts.Throughout, we illustrate the results on some, arti_cial and real, univariateand multivariate data sets.