On the Tukey depth of a continuous probability distribution

Hassairi, Abdelhamid; Regaieg, Ons

doi:10.1016/j.spl.2008.02.008

Cited by 14 publications

(28 citation statements)

References 12 publications

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“…Indeed, by Proposition 31 above, if (33) and (4) are true, then the floating body P [δ] of P exists for all δ ∈ (0, 1/2), and Theorem 34 can be used. Hassairi and Regaieg (2008). Let us state a characterization result for the halfspace depth for absolutely continuous distributions that can be found in [76,Theorem 3.2].…”

Section: Mahalanobis Ellipsoids and The Halfspace Depthmentioning

confidence: 99%

“…Hassairi and Regaieg (2008). Let us state a characterization result for the halfspace depth for absolutely continuous distributions that can be found in [76,Theorem 3.2]. For this, we define for any x ∈ R d the halfspace function…”

Section: Mahalanobis Ellipsoids and The Halfspace Depthmentioning

confidence: 99%

“…In [76, Theorem 3.2], condition (34) is formulated in a slightly different manner in terms of derivatives of functions related to φ x . It is easy to see that for P that satisfies the conditions from Proposition 29,(34) and the corresponding condition from [76] are equivalent.…”

Section: Characterization Theorem Ofmentioning

confidence: 99%

See 2 more Smart Citations

Halfspace depth and floating body

Nagy¹,

Schütt²,

Werner³

2019

Statist. Surv.

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Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

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Section: Mahalanobis Ellipsoids and The Halfspace Depthmentioning

confidence: 99%

Section: Mahalanobis Ellipsoids and The Halfspace Depthmentioning

confidence: 99%

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Halfspace depth and floating body

Nagy¹,

Schütt²,

Werner³

2019

Statist. Surv.

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“…The appealing properties of halfspace depth are well known and well documented: see Donoho and Gasko [14], Mosler [43], Koshevoy [33], Ghosh and Chaudhuri [19], Cuestas-Albertos and Nieto-Reyes [8], Hassairi and Regaieg [29], to cite only a few. Halfspace depth takes values in [0, 1/2], and its contours are continuous and convex; the corresponding regions are closed, convex, and nested as d decreases.…”

Section: Statistical Depth Regions and Contoursmentioning

confidence: 99%

Monge-Kantorovich depth, quantiles, ranks and signs

Henry¹,

Galichon²,

Chernozhukov³

et al. 2015

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We propose new concepts of statistical depth, multivariate quantiles, vector quantiles and ranks, ranks, and signs, based on canonical transportation maps between a distribution of interest on IR d and a reference distribution on the d-dimensional unit ball. The new depth concept, called Monge-Kantorovich depth, specializes to halfspace depth for d = 1 and in the case of spherical distributions, but, for more general distributions, differs from the latter in the ability for its contours to account for non convex features of the distribution of interest. We propose empirical counterparts to the population versions of those Monge-Kantorovich depth contours, quantiles, ranks, signs, and vector quantiles and ranks, and show their consistency by establishing a uniform convergence property for empirical (forward and reverse) transport maps, which is the main theoretical result of this paper.1. Introduction. The concept of statistical depth was introduced in order to overcome the lack of a canonical ordering in IR d for d > 1, hence the absence of the related notions of quantile and distribution functions, ranks, and signs. The earliest and most popular depth concept is halfspace depth, the definition of which goes back to Tukey [54] Zuo and Serfling [62], who list four properties that are generally considered desirable for any statistical depth function, namely affine invariance, maximality at the center, linear monotonicity relative to the deepest points, and vanishing at infinity (see Section 2.2 for details). Halfspace depth is the prototype of a depth concept satisfying the Liu-Zuo-Serfling axioms for the family P

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“…Though the problem is still open in general, a number of promising positive results concerning subclasses of probability distributions have been proved. We mention the work of Struyf and Rousseeuw (1999), Koshevoy (2002), Hassairi and Regaieg (2008), Cuesta-Albertos and Nieto-Reyes (2008) and a recent article by Kong and Zuo (2010). In the latter the authors showed that a probability distribution is characterized by its depth if all the depth contours are smooth.…”

Section: Introductionmentioning

confidence: 98%

On smoothness of Tukey depth contours

Gijbels

Nagy

2016

Statistics

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Abstract. The smoothness of Tukey depth contours is a regularity condition often encountered in asymptotic theory, among others. This condition ensures that the Tukey depth fully characterizes the underlying multivariate probability distribution. In this paper we demonstrate that this regularity condition is rarely satisfied. It is shown that even well-behaved probability distributions with symmetrical, smooth and (strictly) quasiconcave densities may have non-smooth Tukey depth contours, and that the smoothness behaviour of depth contours is fairly unpredictable.

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On the Tukey depth of a continuous probability distribution

Cited by 14 publications

References 12 publications

Halfspace depth and floating body

Halfspace depth and floating body

Monge-Kantorovich depth, quantiles, ranks and signs

On smoothness of Tukey depth contours

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