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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 17 (7 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.
Invariant coordinate selection
, 2007
"... A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based upon the eigenvalueeigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to gener ..."
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Cited by 16 (8 self)
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A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based upon the eigenvalueeigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant coordinate system for the multivariate data. Consequently, we view this method as a method for invariant coordinate selection (ICS). By plotting the data with respect to this new invariant coordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant coordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant coordinates corresponds to Fisher’s linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given.
A canonical definition of shape
, 2007
"... Very general concepts of scatter, extending the traditional notion of covariance matrices, have become classical tools in robust multivariate analysis. In many problems of practical importance (principal components, canonical correlation, testing for sphericity), only homogeneous functions of the ..."
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Cited by 11 (6 self)
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Very general concepts of scatter, extending the traditional notion of covariance matrices, have become classical tools in robust multivariate analysis. In many problems of practical importance (principal components, canonical correlation, testing for sphericity), only homogeneous functions of the scatter matrix are of interest. In line with this fact, scatter functionals often are only defined up to a positive scalar factor, yielding a family of scatter matrices rather than a uniquely defined one. In such families, it is natural to single out one representative by imposing a normalization constraint: this normalized scatter is called a shape matrix. In the particular case of elliptical families, this constraint in turn induces a concept of scale; along with a location center and a standardized radial density, the shape and scale parameters entirely characterize an elliptical density. In this paper, we show that one and only normalization has the additional properties that (i) the resulting Fisher information matrices for shape and scale, in locally asymptotically normal (LAN) elliptical families, are blockdiagonal, and that (ii) the semiparametric elliptical families indexed by location, shape, and completely unspecified radial densities are adaptive. This particular normalization, which imposes that the determinant of the shape matrix be equal to one, therefore can be considered canonical.
OPTIMAL RANKBASED TESTS FOR HOMOGENEITY OF SCATTER
, 806
"... We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly r ..."
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Cited by 8 (8 self)
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We propose a class of locally and asymptotically optimal tests, based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations. Contrary to the existing parametric procedures, these tests remain valid without any moment assumptions, and thus are perfectly robust against heavytailed distributions (validity robustness). Nevertheless, they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all (efficiency robustness). In particular, their normalscore version outperforms traditional Gaussian likelihood ratio tests and their pseudoGaussian robustifications under a very broad range of nonGaussian densities including, for instance, all multivariate Student and powerexponential distributions. 1. Introduction. 1.1. Homogeneity of variances and covariance matrices. The assumption of variance homogeneity is central to the theory and practice of univariate
Optimal tests for homogeneity of covariance, scale, and shape
 J. Multivariate Anal
, 2008
"... The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic cont ..."
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Cited by 7 (4 self)
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The assumption of homogeneity of covariance matrices is the fundamental prerequisite of a number of classical procedures in multivariate analysis. Despite its importance and long history, however, this problem so far has not been completely settled beyond the traditional and highly unrealistic context of multivariate Gaussian models. And the modified likelihood ratio tests (MLRT) that are used in everyday practice are known to be highly sensitive to violations of Gaussian assumptions. In this paper, we provide a complete and systematic study of the problem, and propose test statistics which, while preserving the optimality features of the MLRT under multinormal assumptions, remain valid under unspecified elliptical densities with finite fourthorder moments. As a first step, the Le Cam LAN approach is used for deriving locally and asymptotically optimal testing procedures φ (n) f for any specified mtuple of radial densities f = (f1,..., fm). Combined with an estimation of the m densities f1,..., fm, these procedures can be used to construct adaptive tests for the problem. Adaptive tests however typically require very large samples, and pseudoGaussian tests—namely, tests that are locally and asymptotically optimal at Gaussian densities while remaining valid under a much broader class of distributions—in general are preferable. We therefore construct two pseudoGaussian modifications of the Gaussian version φ (n) N of the optimal test φ (n) f. The first one, φ
A Mahalanobis Multivariate Quantile Function
, 2008
"... The Mahalanobis distance is pervasive throughout multivariate statistical analysis. Here it receives a further role, in formulating a new affine equivariant and mathematically tractable multivariate quantile function with favorable properties. Besides having sample versions that are robust, computat ..."
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Cited by 4 (2 self)
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The Mahalanobis distance is pervasive throughout multivariate statistical analysis. Here it receives a further role, in formulating a new affine equivariant and mathematically tractable multivariate quantile function with favorable properties. Besides having sample versions that are robust, computationally easy, and asymptotically normal, this “Mahalanobis quantile function ” also provides two special benefits. Its associated “outlyingness ” contours, unlike those of the “usual ” Mahalanobis outlyingness function, are not restricted to be elliptical. And it provides a rigorous foundation for understanding the transformationretransformation (TR) method used in practice for constructing affine equivariant versions of the sample spatial quantile function. Indeed, the Mahalanobis quantile function has a TR representation in terms of the spatial quantile function. This yields, for example, that the “TR sample spatial median ” estimates not the population spatial median, but rather the population Mahalanobis median. This clarification actually strengthens, rather than weakens, the motivation for the TR approach. Two major tools, both of independent interest as well, are developed and applied: a general formulation of affine equivariance for multivariate quantile functions, and a notion of “weak covariance functional ” that connects with the functionals used in the TR approach. Variations on the definition of “Mahalanobis ” quantiles are also discussed.
Multivariate Generalized Sestimators
, 2008
"... In this paper we introduce generalized Sestimators for the multivariate regression model. This class of estimators combines high robustness and high efficiency. They are defined by minimizing the determinant of a robust estimator of the scatter matrix of differences of residuals. In the special cas ..."
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Cited by 4 (0 self)
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In this paper we introduce generalized Sestimators for the multivariate regression model. This class of estimators combines high robustness and high efficiency. They are defined by minimizing the determinant of a robust estimator of the scatter matrix of differences of residuals. In the special case of a multivariate location model, the generalized Sestimator has the important independence property, and can be used for high breakdown estimation in independent component analysis. Robustness properties of the estimators are investigated by deriving their breakdown point and the influence function. We also study the efficiency of the estimators, both asymptotically and at finite samples. To obtain inference for the regression parameters, we discuss the fast and robust bootstrap for multivariate generalized
Spatial trimming, with applications to robustify sample spatial quantile and outlyingness functions, and to construct a new robust scatter estimator. submitted
, 2010
"... The spatial multivariate median has a long history as an alternative to the sample mean. Its transformationretransformation (TR) sample version is affine equivariant, highly robust, and computationally easy. More recently, an entire TR spatial multivariate quantile function has been developed and a ..."
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Cited by 2 (2 self)
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The spatial multivariate median has a long history as an alternative to the sample mean. Its transformationretransformation (TR) sample version is affine equivariant, highly robust, and computationally easy. More recently, an entire TR spatial multivariate quantile function has been developed and applied in practice along with related rank functions. However, as quantile levels move farther out, robustness of the TR sample version as measured by breakdown point decreases to zero, a serious limitation in applications such as outlier detection and setting inner 50%, 75%, and 90 % quantile regions. Here we introduce a new device, “spatial trimming”, and with it solve two problems of general scope and application: (i) the need for robustification of the TR sample spatial quantile function and its closely related depth, outlyingness, and rank functions, and (ii) the need for a computationally easy, robust, and affine equivariant scatter estimator. Improvements in robustness accomplished by spatial trimming are confirmed by improved breakdown points and illustrated using simulated and actual data. Other applications of spatial trimming are
Some facts about functionals of location and scatter
, 2006
"... Abstract: Assumptions on a likelihood function, including a local GlivenkoCantelli condition, imply the existence of Mestimators converging to an Mfunctional. Scatter matrixvalued estimators, defined on all empirical measures on R d for d ≥ 2, and equivariant under all, including singular, affin ..."
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Cited by 1 (1 self)
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Abstract: Assumptions on a likelihood function, including a local GlivenkoCantelli condition, imply the existence of Mestimators converging to an Mfunctional. Scatter matrixvalued estimators, defined on all empirical measures on R d for d ≥ 2, and equivariant under all, including singular, affine transformations, are shown to be constants times the sample covariance matrix. So, if weakly continuous, they must be identically 0. Results are stated on existence and differentiability of location and scatter functionals, defined on a weakly dense, weakly open set of laws, via elliptically symmetric t distributions on R d, following up on work of Kent, Tyler, and Dümbgen. 1.
On Multivariate Runs Tests for Randomness
"... matrix This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999 ..."
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Cited by 1 (1 self)
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matrix This paper proposes several extensions of the concept of runs to the multivariate setup, and studies the resulting tests of multivariate randomness against serial dependence. Two types of multivariate runs are defined: (i) an elliptical extension of the spherical runs proposed by Marden (1999), and (ii) an original concept of matrixvalued runs. The resulting runs tests themselves exist in various versions, either based on spatial signs (see, e.g., Möttönen and Oja 1995, Randles 2000) or on the hyperplanebased multivariate signs known as interdirections (see, e.g., Randles 1989, Taskinen, Oja, and Randles 2005). All proposed multivariate runs tests are affineinvariant and highly robust: in particular, they allow for heteroskedasticity and do not require any moment assumption. Their limiting distributions are derived under the null hypothesis and under sequences of local vector ARMA alternatives. Asymptotic relative efficiencies with respect to Gaussian Portmanteau tests are computed, and show that, while Mardentype runs tests suffer severe consistency problems, tests based on matrixvalued runs perform uniformly well for moderatetolarge dimensions. A MonteCarlo study confirms the theoretical results and investigates the robustness properties of the proposed procedures. A real data example is treated, and shows that combining Mardentype runs tests and tests based on matrixvalued runs may provide some insight on the reason why rejection occurs.