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Theory of multivariate statistics
, 1999
"... Our object in writing this book is to present the main results of the modern theory of multivariate statistics to an audience of advanced students who would appreciate a concise and mathematically rigorous treatment of that material. It is intended for use as a textbook by students taking a first gr ..."
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Our object in writing this book is to present the main results of the modern theory of multivariate statistics to an audience of advanced students who would appreciate a concise and mathematically rigorous treatment of that material. It is intended for use as a textbook by students taking a first graduate course in the subject, as well as for the general reference of interested research workers who will find, in a readable form, developments from recently published work on certain broad topics not otherwise easily accessible, as, for instance, robust inference (using adjusted likelihood ratio tests) and the use of the bootstrap in a multivariate setting. The references contains over 150 entries post1982. The main development of the text is supplemented by over 135 problems, most of which are original with the authors. A minimum background expected of the reader would include at least two courses in mathematical statistics, and certainly some exposure to the calculus of several variables together with the descriptive geometry of linear
Semiparametrically efficient rankbased inference for shape I: Optimal rankbased tests for sphericity
 Ann. Statist
, 2006
"... A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under a ..."
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Cited by 16 (12 self)
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A class of Restimators based on the concepts of multivariate signed ranks and the optimal rankbased tests developed in Hallin and Paindaveine [Ann. Statist. 34 (2006)] is proposed for the estimation of the shape matrix of an elliptical distribution. These Restimators are rootn consistent under any radial density g, without any moment assumptions, and semiparametrically efficient at some prespecified density f. When based on normal scores, they are uniformly more efficient than the traditional normaltheory estimator based on empirical covariance matrices (the asymptotic normality of which, moreover, requires finite moments of order four), irrespective of the actual underlying elliptical density. They rely on an original rankbased version of Le Cam’s onestep methodology which avoids the unpleasant nonparametric estimation of crossinformation quantities that is generally required in the context of Restimation. Although they are not strictly affineequivariant, they are shown to be equivariant in a weak asymptotic sense. Simulations confirm their feasibility and excellent finitesample performances. 1. Introduction. 1.1. Rankbased inference for elliptical families. An elliptical density over Rk is determined by a location center θ ∈ Rk, a scale parameter σ ∈ R + 0, a realvalued positive definite symmetric k × k matrix V = (Vij) with V11 = 1,
The distribution of robust distances
 Journal of Computational and Graphical Statistics
, 2005
"... Mahalanobistype distances in which the shape matrix is derived from a consistent, highbreakdown robust multivariate location and scale estimator have an asymptotic chisquared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw’s minimum covari ..."
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Mahalanobistype distances in which the shape matrix is derived from a consistent, highbreakdown robust multivariate location and scale estimator have an asymptotic chisquared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw’s minimum covariance determinant (MCD) is a robust estimator with a high breakdown. However, even in quite large samples, the chisquared approximation to the distances of the sample data from the MCD center with respect to the MCD shape is poor. We provide an improved F approximation that gives accurate outlier rejection points for various sample sizes.
Robust Multivariate Regression
 Technometrics
, 2000
"... We introduce a robust method for multivariate regression, based on robust estimation of the joint location and scatter matrix of the explanatory and response variables. The resulting method has the appropriate equivariance properties, a bounded inuence function, and the same breakdown value as the i ..."
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Cited by 9 (2 self)
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We introduce a robust method for multivariate regression, based on robust estimation of the joint location and scatter matrix of the explanatory and response variables. The resulting method has the appropriate equivariance properties, a bounded inuence function, and the same breakdown value as the initial estimators of location and scatter. To increase the efficiency we propose a reweighted estimator, which was selected from several possible reweighting schemes. Simulations show that the asymptotic properties of robustness and efficiency remain valid at finite samples. The method does not need much computation time, and is applied to chemical engineering data. Key words: breakdown value; distancedistance plot; inuence function; minimum covariance determinant; reweighting. Peter J. Rousseeuw is Professor and Stefan Van Aelst is Research Assistant with the FWO Belgium, both at the Department of Mathematics and Computer Science, University of Antwerp (UIA), Universiteitsplein 1, B2610...
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 4 (4 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
Asymptotic expansion of Sestimators of location and covariance
, 1995
"... By means of an elementary application of empirical process theory, we show that Sestimators of multivariate location and covariance are asymptotically equivalent to a sum of independent vector and matrix valued random elements respectively. This provides an alternative proof of asymptotic normality ..."
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By means of an elementary application of empirical process theory, we show that Sestimators of multivariate location and covariance are asymptotically equivalent to a sum of independent vector and matrix valued random elements respectively. This provides an alternative proof of asymptotic normality of Sestimators and clearly explains the limiting covariance structure. It also leads to a relatively simple proof of asymptotic normality of the length of the shortest fffraction.
Asymptotic covariance matrices of different shape matrix estimators. Discussion paper
, 2007
"... 4 th version ..."
Robust Methods for Mean and Covariance Structure Analysis
, 1995
"... Covariance structure analysis plays an important role in social and behavioral sciences to evaluate hypothesized influences among unmeasured latent and observed variables. Existing methods for analyzing these data rely on unstructured sample means and covariances estimated under normality, and evalu ..."
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Covariance structure analysis plays an important role in social and behavioral sciences to evaluate hypothesized influences among unmeasured latent and observed variables. Existing methods for analyzing these data rely on unstructured sample means and covariances estimated under normality, and evaluate a proposed structural model using statistical theory based on normal theory MLE and generalized least squares (GLS) with a weight matrix obtained from inverting a matrix based on sample fourth moments and covariances. Since the influence functions associated with these methods are quadratic, a few outliers can make these classical procedures a total failure. Considering that data collected in social and behavioral sciences are not so accurate, some robust methods are necessary in estimation and testing. Even though the theory for robustly estimating multivariate location and scatter has been developed extensively, very little has been accomplished in robust mean and covariance structure ...
Robust estimation of the SUR model
"... This paper proposes robust regression to solve the problem of outliers in seemingly unrelated regression (SUR) models. The authors present an adaptation of Sestimators to SUR models. Sestimators are robust, with high breakdown point, and are much more e#cient than other robust regression estimator ..."
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This paper proposes robust regression to solve the problem of outliers in seemingly unrelated regression (SUR) models. The authors present an adaptation of Sestimators to SUR models. Sestimators are robust, with high breakdown point, and are much more e#cient than other robust regression estimators commonly used in practice. Furthermore, modifications to Ruppert's algorithm allow a fast evaluation of them in this context. The classical example of U.S. corporations is revisited, and it appears that the procedure gives an interesting insight into the problem. R ESUM E Les auteurs proposent une methode de regression robuste pour resoudre le probleme des valeurs aberrantes dans les modeles SUR. Ils adaptent les Sestimateurs dans les modeles SUR. Les Sestimateurs sont robustes, avec un haut point de rupture, et sont beaucoup plus e#caces que les autres estimateurs robustes de regression couramment utilises en pratique. De plus, une modification de l'algorithme de Ruppert permet une ev...
A Unified Approach to Multigroup Structural Equation Modeling with Nonstandard Samples
"... Introduction It is well known that structural equation modeling (SEM) has become one of the most popular methods in multivariate analysis, especially in the social and behavioral sciences. In a SEM model with latent variables, the relationships among observed (manifest) variables is formulated throu ..."
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Introduction It is well known that structural equation modeling (SEM) has become one of the most popular methods in multivariate analysis, especially in the social and behavioral sciences. In a SEM model with latent variables, the relationships among observed (manifest) variables is formulated through unobserved (latent) constructs. Because measurement errors are explicitly accounted for, coe#cients in key parts of a model are uninfluenced by errors of measurement, implying greater theoretical meaningfulness and crosspopulation stability to the parameters than might be achieved with methods such as regression or analysis of variance that do not correct for unreliability. This stability is a key goal of theory testing with SEM, where a substantive theory or hypothesized causal relationship among the latent constructs, facilitated by path diagrams, can be tested through SEM. With the help of popular software such as LISREL (Joreskog & Sorbom, 1993) and EQS (Bentler, 2000), applications