Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean

Massé, Jean–Claude

doi:10.3150/bj/1089206404

Cited by 47 publications

(54 citation statements)

References 18 publications

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“…The asymptotic properties of D n (x) are studied in several papers including those of Liu [1990], Arcones et al [1994], Massé [2004] and Lin and Chen [2006]. Liu and Singh [1993] established for the sample Mahalanobis depth function that sup x jD n (x) À D(x; F)j converges to zero almost surely as n goes to infinity, under suitable conditions on F. For convenience, the following notation is used for the Mahalanobis depth function in the next sections:…”

Section: Depth Functionmentioning

confidence: 99%

Depth and homogeneity in regional flood frequency analysis

Chebana

Ouarda

2008

Water Resources Research

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[1] Regional frequency analysis (RFA) consists generally of two steps: (1) delineation of hydrological homogeneous regions and (2) regional estimation. Existing regionalization methods which adopt this two-step approach suffer from two principal drawbacks. First, the restriction of the regional estimation to a particular region by excluding some sites can correspond to a loss of some information. Second, the definition of a region generates a border effect problem. To overcome these problems, a new method is proposed in the present paper. The proposed method is based on three elements: (1) a weight function to treat the border effect problem, (b) a function to evaluate how ''similar'' each site is to the target one, and (c) an iterative procedure to improve estimation results. Element (b) is treated using the statistical notion of depth functions which is introduced to provide a ranking of stations in a multivariate context. Furthermore, the properties of depth functions meet the characteristics sought in RFA. It is shown that the proposed method is flexible and general and that traditional RFA methods represent special cases of the depth-based approach corresponding to particular weight functions. A comparison is carried out with the canonical correlation analysis (CCA) approach. Results indicate that the depth-based approach performs better than does CCA both in terms of relative bias and relative root mean squares error.

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Section: Depth Functionmentioning

confidence: 99%

Depth and homogeneity in regional flood frequency analysis

Chebana

Ouarda

2008

Water Resources Research

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“…From the definition of α , we conclude that D(x) = α . Then according to a result due to Massé (2004) implies that ∂ H [x, v] is a hyperplane of support of Q α at x.…”

Section: Resultsmentioning

confidence: 94%

“…Then according to (H), the hyperplane ∆ is minimal at one of its points. This implies that ∆ is a hyperplane of support of some Q α (see Massé (2004)). …”

Section: Resultsmentioning

confidence: 97%

On the Tukey depth of a continuous probability distribution

Hassairi

Regaieg

2008

Statistics & Probability Letters

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“…We first introduce some facts concerning the Tukey depth; our notations are the ones used in [4]. Let |x| denote the Euclidean norm of an element x of R d , x, y the scalar product of x and y, and let…”

Section: Resultsmentioning

confidence: 99%

“…Roughly speaking, one can think of points with high depth as being close to the "center" of the distribution and points with low depth as belonging to the tails (see [4]). One of the most famous among these notions is the half-space depth function introduced by Tukey [8].…”

Section: Introductionmentioning

confidence: 99%