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45
Structural Properties and Convergence Results for Contours of Sample Statistical Depth Functions
, 2000
"... Statistical depth functions have become increasingly used in nonparametric inference for multivariate data. Here the contours of such functions are studied. Structural properties of the regions enclosed by contours, such as affine equivariance, nestedness, connectedness, and compactness, and almost ..."
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Cited by 22 (12 self)
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Statistical depth functions have become increasingly used in nonparametric inference for multivariate data. Here the contours of such functions are studied. Structural properties of the regions enclosed by contours, such as affine equivariance, nestedness, connectedness, and compactness, and almost sure convergence results for sample depth contours, are established. Also, specialized results are established for some popular depth functions, including halfspace depth, and for the case of elliptical distributions. Finally, some needed foundational results on almost sure convergence of sample depth functions are provided.
Quantile Functions for Multivariate Analysis: Approaches and Applications
, 2001
"... Despite the absence of a natural ordering of Euclidean space for dimension greater than one, effort to define vectorvalued quantile functions for multivariate distributions has generated several approaches. To support greater discrimination in comparing, selecting and using such functions, we in ..."
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Cited by 19 (4 self)
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Despite the absence of a natural ordering of Euclidean space for dimension greater than one, effort to define vectorvalued quantile functions for multivariate distributions has generated several approaches. To support greater discrimination in comparing, selecting and using such functions, we introduce relevant criteria, including a notion of "medianoriented quantile function". On this basis we compare recent quantile approaches and several multivariate versions of trimmed mean and interquartile range. We also discuss a univariate "generalized quantile" approach that enables particular features of multivariate distributions, for example scale and kurtosis, to be studied by twodimensional plots. Methods based on statistical depth functions are found to be especially attractive for quantilebased multivariate inference.
On the Performance of Some Robust Nonparametric Location Measures Relative to a General Notion of Multivariate Symmetry
 J. STATIST. PLANN. INFERENCE
, 1999
"... Several robust nonparametric location estimators are examined with respect to several criteria, with emphasis on the criterion that they should agree with the point of symmetry in the case of a symmetric distribution. For this purpose, a broad version of multidimensional symmetry is introduced, name ..."
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Cited by 16 (9 self)
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Several robust nonparametric location estimators are examined with respect to several criteria, with emphasis on the criterion that they should agree with the point of symmetry in the case of a symmetric distribution. For this purpose, a broad version of multidimensional symmetry is introduced, namely "halfspace symmetry", generalizing the wellknown notions of "central" and "angular" symmetry. Characterizations of these symmetry notions are established, permitting their properties and interrelations to be illuminated. The particular location measures considered consist of several nonparametric notions of multidimensional median: the "L²" (or "spatial"), "Tukey/Donoho halfspace", "projection", and "Liu simplicial" medians, all of which are robust in the sense of nonzero breakdown point. It is established that the rst three of these in general do identify the point of symmetry when it exists, whereas the latter, however, fails to do so in some circumstances. Combining this finding ...
Lower Bounds for Computing Statistical Depth
, 2001
"... Given a finite set of points S, two measures of the depth of a query point with respect to S are the Simplicial depth of Liu and the Halfspace depth of Tukey (also known as Location depth). We show that computing these depths requires n log n) time, which matches the upper bound complexities o ..."
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Cited by 12 (4 self)
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Given a finite set of points S, two measures of the depth of a query point with respect to S are the Simplicial depth of Liu and the Halfspace depth of Tukey (also known as Location depth). We show that computing these depths requires n log n) time, which matches the upper bound complexities of the algorithms of Rousseeuw and Ruts. Our lower bound proofs may also be applied to two bivariate sign tests: that of Hodges, and that of Oja and Nyblom.
Nonparametric Notions of Multivariate "Scatter Measure" and "More Scattered" Based on Statistical Depth Functions
, 1999
"... Nonparametric notions of multivariate "scatter measure" and "more scattered", based on statistical depth functions, are investigated. In particular, notions of "more scattered" based on the "halfspace" depth function are shown to generalize versions introduced by Bickel and Lehmann (1976, 1979) in t ..."
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Cited by 10 (8 self)
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Nonparametric notions of multivariate "scatter measure" and "more scattered", based on statistical depth functions, are investigated. In particular, notions of "more scattered" based on the "halfspace" depth function are shown to generalize versions introduced by Bickel and Lehmann (1976, 1979) in the univariate case and by Eaton (1982) and Oja (1983) in the multivariate case. Scatter measures are also discussed, with emphasis on those based on the halfspace depth. Basic desirable properties established for the previous versions of "more scattered" are shown to carry over to the depthbased notions as well, in both the univariate and multivariate cases. Further, some properties unique to the depthbased notions are established. AMS 1991 Subject Classification: Primary 62H05; Secondary 62G05. Key words and phrases: Scatter measure; more scattered; statistical depth functions; multivariate; nonparametric; statistical inference; halfspace depth. 1 Introduction Statistical depth funct...
On the StahelDonoho estimator and depthweighted means of multivariate data
 Annals of Statistics
, 2004
"... Depth of multivariate data can be used to construct weighted means as robust estimators of location. The use of projection depth leads to the StahelDonoho estimator as a special case. In contrast to maximal depth estimators, the depthweighted means are shown to be asymptotically normal under appr ..."
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Cited by 10 (2 self)
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Depth of multivariate data can be used to construct weighted means as robust estimators of location. The use of projection depth leads to the StahelDonoho estimator as a special case. In contrast to maximal depth estimators, the depthweighted means are shown to be asymptotically normal under appropriate conditions met by depth functions commonly used in the current literature. We also confirm through a finitesample study that the StahelDonoho estimator achieves a desirable balance between robustness and efficiency at Gaussian models.
Equivariance and Invariance Properties of Multivariate Quantile and Related Functions, and the Role of Standardization
, 2009
"... Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invar ..."
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Cited by 9 (6 self)
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Equivariance and invariance issues arise as a fundamental but often problematic aspect of multivariate statistical analysis. For multivariate quantile and related functions, we provide coherent definitions of these properties. For standardization of multivariate data to produce equivariance or invariance of procedures, three important types of matrixvalued functional are studied: “weak covariance ” (or “shape”), “transformationretransformation ” (TR), and “strong invariant coordinate system ” (SICS). Clarification of TR affine equivariant versions of the sample spatial quantile function is obtained. It is seen that geometric artifacts of SICSstandardized data are invariant under affine transformation of the original data followed by standardization using the same SICS functional, subject only to translation and homogeneous scale change. Some applications of SICS standardization are described.
Generalized Quantile Processes Based on Multivariate Depth Functions, with Applications in Nonparametric Multivariate Analysis
, 2001
"... Statistical depth functions are being used increasingly in nonparametric multivariate data analysis. In a broad treatment of depthbased methods, Liu, Parelius and Singh (1999) include several devices for visualizing selected multivariate distributional characteristics by onedimensional curves cons ..."
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Cited by 8 (5 self)
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Statistical depth functions are being used increasingly in nonparametric multivariate data analysis. In a broad treatment of depthbased methods, Liu, Parelius and Singh (1999) include several devices for visualizing selected multivariate distributional characteristics by onedimensional curves constructed in terms of given depth functions. Here we show how these tools may be represented as special depthbased cases of generalized quantile functions introduced by Einmahl and Mason (1992). By specializing results of the latter authors to the depthbased case, we develop an easily applied general result on convergence of sample depthbased generalized quantile processes to a Brownian bridge. As applications, we obtain the asymptotic behavior of sample versions of depthbased curves for "scale" and "kurtosis" introduced by Liu, Parelius and Singh. The kurtosis curve is actually a Lorenz curve designed to measure heaviness of tails of a multivariate distribution. We also obtain the asymptotic distribution of the quantile process of the sample depth values.
Nonparametric depthbased multivariate outlier identifiers, and robustness properties
, 2006
"... DMS0103698 and CCF0430366 is gratefully acknowledged. In extending univariate outlier detection methods to higher dimension, various special issues arise, such as limitations of visualization methods, inadequacy of marginal methods, lack of a natural order, limited scope of parametric modeling, an ..."
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Cited by 8 (5 self)
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DMS0103698 and CCF0430366 is gratefully acknowledged. In extending univariate outlier detection methods to higher dimension, various special issues arise, such as limitations of visualization methods, inadequacy of marginal methods, lack of a natural order, limited scope of parametric modeling, and restriction to ellipsoidal contours when using Mahalanobis distance methods. Here we pass beyond these limitations via an approach based on depth functions, which order multidimensional data points by “outlyingness ” measures and generate contours following the shape of the data set. This approach to multivariate outlier detection is nonparametric and, with typical choices of depth function, robust. For depthbased outlier identifiers, we define masking and swamping breakdown points, adapting ideas of Davies
A depth function and a scale curve based on spatial quantiles
 In Statistical Data Analysis Based On the L1Norm and Related Methods
, 2002
"... Abstract. Spatial quantiles, based on the L1 norm in a certain sense, provide an appealing vector extension of univariate quantiles and generate a useful “volume ” functional based on spatial “central regions ” of increasing size. A plot of this functional as a “spatial scale curve ” provides a conv ..."
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Cited by 8 (6 self)
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Abstract. Spatial quantiles, based on the L1 norm in a certain sense, provide an appealing vector extension of univariate quantiles and generate a useful “volume ” functional based on spatial “central regions ” of increasing size. A plot of this functional as a “spatial scale curve ” provides a convenient twodimensional characterization of the spread of a multivariate distribution of any dimension. We discuss this curve and establish weak convergence of the empirical version. As a tool, we introduce and study a new statistical depth function which is naturally associated with spatial quantiles. Other depth functions that generate L1based multivariate quantiles are also noted. 1.