Projection-based depth functions and associated medians

By Yijun Zuo. 2003

324 citations recorded by the source corpus.

Subfields: Electrical and Electronic Engineering, Mechanical Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/91a978640a71ece21d348ec3611d58575ba71d0f

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Citations in the corpus

On the Concept of Depth for Functional Data (2009, methodology): “Several depth definitions for multivariate data have been proposed and analyzed by Mahalanobis (1936), Tukey (1975), Oja (1983), Liu (1990), Singh (1991), Fraiman and Meloche 1999), Vardi and Zhang (2000), Koshevoy and Mosler (1997) and Zuo (2003), among others.”
Excursion probability of Gaussian random fields on sphere (2014, methodology): “This result allows one to construct confidence regions for the true projection median defined in Zuo ([45], Section 3) without using the bootstrapping techniques.”
On the limiting distributions of multivariate depth-based rank sum statistics and related tests (2006, background): “3 of [25] and (A3) follows from (C3) and the fact that for any x ∈ R and some constant C > 0, ∣∣PD(x;Fm) − PD(x;F)∣∣ ≤ sup u∈Sd−1 O(x;F)|σ(Fmu) − σ(Fu)| + |μ(Fmu) − μ(Fu)| (1 + O(x;Fm))(1 + O(x;F))σ (Fmu) ≤ C sup u∈Sd−1 {|σ(Fmu) − σ(Fu)| + |μ(Fmu) − μ(Fu)|}.”
Some intriguing properties of Tukey's half-space depth (2012, unknown): “median based on projection depth (see [23]).”
Robust Estimators are Hard to Compute (2006, background): “The following two definitions can be found in [16]:”
Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation (2010, background): “For the set of all projections, this is the usual projection outlyingness (see e.g. Zuo 2003) and is affine invariant.”
Choosing Among Notions of Multivariate Depth Statistics (2020, background): “Another indicator of robustness of a depth is the influence function of its median; it is investigated in Romanazzi (2001) for halfspace depth, in Niinimaa and Oja (1995) for L2-, spatial, and Oja depth, in Zuo (2004) for Lp depth, and in Zuo (2006) for projection depth.”
Multivariate spacings based on data depth: I. Construction of nonparametric multivariate tolerance regions (2008, background): “Existing notions of data depth include: Mahalanobis depth ([20]), half-space depth ([14, 25]), simplicial depth ([16]), projection depth ([7, 8, 23, 30]), etc.”
Robust estimation and classification for functional data via projection-based depth notions (2007, methodology): “A projection-based method is proposed in Zuo (2003) ; see also Zuo and Serfling (2000).”
Depth weighted scatter estimators (2005, methodology): “Conditions (B1) and (B2) hold for common choices of (µ,σ) and a wide range of distributions; see Remark 2.4 of Zuo (2003) for a detailed discussion [also see Zuo, Cui and He (2004)].”
Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels (2008, background): “Some well-known examples are the halfspace depth (Tukey, 1975), the Oja’s depth (Oja, 1983), the simplicial depth (Liu, 1990), and the projection depth (Zuo, 2003).”
Multidimensional trimming based on projection depth (2006, methodology): “ture; see [16], for example. For any given notion, one can deﬁne and study the corresponding depth trimmed means. Among the existing notions, the projection depth represents a very favorable one; see [33, 34, 37]. Tukey halfspace depth, also built based on projection pursuit methodology, is its majo”
Breakdown and groups (2005, background): “Wiley, New York. HUBERT, M. (1997). The breakdown value of the L1 estimator in contingency tables.”
Multivariate Functional Halfspace Depth (2014, background): “It is however well known that other depth functions, such as projection depth (Zuo, 2003) and spatial depth (Serfling, 2002), attain a higher breakdown value and thus could lead to a more robust MFD.”
Data depths satisfying the projection property (2004, background): “This class of depths generalizes the class of projection depths considered in Zuo (2003) .”