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.

Detecting a structural change in functional time series using local Wilcoxon statistic

“…It ranges from a very misty comparison (parameter close to one) to a very sharp comparison (the parameter close to zero). From other point of view we can treat the statistic (5) as an aggregate representing local asymmetry in data set (see (Paindaveine and Van Bever, 2013)). Differences in value of (5) We expect significant differences in sums of ranks for samples drawn from different continuous distributions however (different distributions should be characterized by different kinds of local asymmetry).…”

“…Such extensions are available in the literature, under the name local depths. In this paper we use the concept of local depth proposed by Paindaveine and Van Bever (2013) and implemented among others in Kosiorowski and Zawadzki (2014). Thorough presentation of the depth concept may be found in (Zuo and Serfling, 2000;Mosler, 2013;Nieto-Reyes and Battey, 2016 …”

“…When defining local depth, following the concept of Paindaveine and Van Bever (2013), it will be more appropriate to index the family {R α (P )} by means of probability contents. Consequently, for any β ∈ (0, 1] we define the smallest depth region with P-probability equal or larger than β as…”

“…The underlying idea is to compare populations at different locality level, which may be interpreted as data resolution. In our proposal the local Wilcoxon test statistic is induced by the corrected modified band depth (López-Pintado and Romo, 2007) with a concept of locality proposed by (Paindaveine and Van Bever, 2013). The rest of the paper is organized as follows.…”

“…Flores et al (2015)). We propose a novel statistic for detetecting a structural change in functional time series based on a local Wilcoxon statistic induced by a local depth function proposed in Paindaveine and Van Bever (2013). …”