A practical affine equivariant multivariate median

Hettmansperger, Thomas P.; Randles, Ronald H.

doi:10.1093/biomet/89.4.851

Cited by 144 publications

(111 citation statements)

References 29 publications

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“…Classical choices for S include S( ) = 11 ([6], [5], [9], and [16]), S( ) = (tr )/k ( [4], [13], and [20]), and S( ) = | | 1/k ( [3], [17], [18], and [19]). This leads to rewriting (1.1) as Irrespective of the choice of S, this shape matrix V S is a normalized version of the scatter matrix -hence, under finite second-order moments, a normalized version of the ordinary covariance matrix.…”

Section: Scatter Scale and Shapementioning

confidence: 99%

Parametric and semiparametric inference for shape: the role of the scale functional

Hallin¹,

Paindaveine²

2006

Statistics &Amp; Decisions

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Summary:We are considering the problem of efficient inference on the shape matrix of an elliptic distribution with unspecified location and either (a) fully specified radial density, (b) radial density specified up to a scale parameter, or (c) completely unspecified radial density. Bickel in [1] has shown that efficiencies under (b) and (c), while being strictly less than under (a), coincide: the efficiency loss caused by an unspecified radial density thus is entirely due to the non-specification of its scale (scale here is not necessarily measured by standard error, as second-order moments may be infinite). Defining scale however requires the choice of a particular scale functional, a choice which has an impact on efficiency bounds. We provide a closed form expression for this efficiency loss, both in hypothesis testing and in point estimation, as a function of the standardized radial density and the scale functional. We show that this loss is maximum at arbitrarily light tails whereas, under arbitrarily heavy tails, it is arbitrarily close to zero: hence, under very heavy tails, ignoring the scale (ignoring the exact density) asymptotically does not harm inference on shape. However, the same loss is nil, irrespective of the standardized radial density, when the scale functional (in dimension k) is the k-th root of the scatter determinant.

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Section: Scatter Scale and Shapementioning

confidence: 99%

Parametric and semiparametric inference for shape: the role of the scale functional

Hallin¹,

Paindaveine²

2006

Statistics &Amp; Decisions

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“…Note also that, if the standardization is done using such location vector and scatter (or shape) matrix estimates that do not require any moment assumptions of the underlying data (e.g. Tyler's shape matrix and the transformation retransformation spatial median in Hettmansperger and Randles, 2002), then the resulting test procedures are valid without any moment assumptions. i be constructed as before.…”

Section: The Rank Scores Test Statisticsmentioning

confidence: 99%

Rank Scores Tests of Multivariate Independence

Taskinen

Kankainen

Oja

2004

Theory and Applications of Recent Robust Methods

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Abstract. New rank scores test statistics are proposed for testing whether two random vectors are independent. The tests are asymptotically distributionfree for elliptically symmetric marginal distributions. Recently, Gieser and Randles (1997), Taskinen, Kankainen and Oja (2003) and Taskinen, Oja and Randles (2005) introduced and discussed different multivariate extensions of the quadrant test, Kendall's tau and Spearman's rho statistics. In this paper, standardized multivariate spatial signs and the (univariate) ranks of the Mahalanobis-type distances of the observations from the origin are combined to construct rank scores tests of independence. The limiting distributions of the test statistics are derived under the null hypothesis as well as under contiguous sequences of alternatives. Three different choices of the score functions, namely the sign scores, the Wilcoxon scores and the van der Waerden scores, are discussed in greater detail. The small sample and limiting efficiencies of the test procedures are compared and the robustness properties are illustrated by an example. It is remarkable that, in the multinormal case, the limiting Pitman efficiency of the van der Waerden scores test equals to that of the classical parametric Wilks' test.

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“…An immense amount of research related to L 1 -median has been carried out (see for example Hayford, 1902;Haldane, 1948;Brown, 1983;Pollard, 1984;Rao, 1988;Small, 1990;Chaudhuri, 1992;Chakraborty, Chaudhuri and Oja, 1998;Dodge and Rousson, 1999;and Hettmansperger and Randles;2002). The uniqueness of the L 1 median is proved by Milasevic and Ducharme (1987), among others.…”

Section: F ) Is Fisher Consistent For Symmetric F and Convex W; (4) Wmentioning

confidence: 99%

Robustness of weighted L p ?depth and L p ?median

Zuo

2004

Allgemeines Statistisches Arch

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Summary: L p -norm weighted depth functions are introduced and the local and global robustness of these weighted L p -depth functions and their induced multivariate medians are investigated via influence function and finite sample breakdown point. To study the global robustness of depth functions, a notion of finite sample breakdown point is introduced. The weighted L p -depth functions turn out to have the same low breakdown point as some other popular depth functions. Their influence functions are also unbounded. On the other hand, the weighted L p -depth induced medians are globally robust with the highest possible breakdown point for any reasonable estimator. The weighted L p -medians are also locally robust with bounded influence functions for suitable weight functions. Unlike other existing depth functions and multivariate medians, the weighted L p depth and medians are easy to calculate in high dimensions. The price for this advantage is the lack of affine invariance and equivariance of the weighted L p depth and medians, respectively.

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A practical affine equivariant multivariate median

Cited by 144 publications

References 29 publications

Parametric and semiparametric inference for shape: the role of the scale functional

Parametric and semiparametric inference for shape: the role of the scale functional

Rank Scores Tests of Multivariate Independence

Robustness of weighted L p ?depth and L p ?median

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