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39
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 48 (32 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,
Analysis of covariance structures under elliptical distributions
 Journal of the American Statistical Association
, 1987
"... This article examines the adjustment of normal theory methods for the analysis of covariance structures to make them applicable under the class of elliptical distributions. It is shown that if the model satisfies a mild scale invariance condition and the data have an elliptical distribution, the asy ..."
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Cited by 22 (0 self)
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This article examines the adjustment of normal theory methods for the analysis of covariance structures to make them applicable under the class of elliptical distributions. It is shown that if the model satisfies a mild scale invariance condition and the data have an elliptical distribution, the asymptotic covariance matrix of sample covariances has a structure that results in the retention of many of the asymptotic properties of normal theory methods. If a scale adjustment is applied, the likelihood ratio tests of fit have the usual asymptotic chisquared distributions. Difference tests retain their property of asymptotic independence, and maximum likelihood estimators retain their relative asymptotic efficiency within the class of estimators based on the sample covariance matrix. An adjustment to the asymptotic covariance matrix of normal theory maximum likelihood estimators for elliptical distributions is provided. This adjustment is particularly simple in models for patterned covariance or correlation matrices. These results apply not only to normal theory maximum likelihood methods but also to a class of minimum discrepancy methods. Similar results also apply when certain robust estimators of the covariance matrix are employed.
Spectral Methods for Learning Multivariate Latent Tree Structure
"... This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linearGaussian models, hidden Markov models, Gaussian mixture models, and Markov evol ..."
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Cited by 18 (4 self)
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This work considers the problem of learning the structure of multivariate linear tree models, which include a variety of directed tree graphical models with continuous, discrete, and mixed latent variables such as linearGaussian models, hidden Markov models, Gaussian mixture models, and Markov evolutionary trees. The setting is one where we only have samples from certain observed variables in the tree, and our goal is to estimate the tree structure (i.e., the graph of how the underlying hidden variables are connected to each other and to the observed variables). We propose the Spectral Recursive Grouping algorithm, an efficient and simple bottomup procedure for recovering the tree structure from independent samples of the observed variables. Our finite sample size bounds for exact recovery of the tree structure reveal certain natural dependencies on underlying statistical and structural properties of the underlying joint distribution. Furthermore, our sample complexity guarantees have no explicit dependence on the dimensionality of the observed variables, making the algorithm applicable to many highdimensional settings. At the heart of our algorithm is a spectral quartet test for determining the relative topology of a quartet of variables from secondorder statistics. 1
OPTIMAL RANKBASED TESTING FOR PRINCIPAL COMPONENTS
"... This paper provides parametric and rankbased optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudoGaussian robustifications by Tyler (1981, 1983) a ..."
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Cited by 13 (11 self)
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This paper provides parametric and rankbased optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families. The parametric tests extend the Gaussian likelihood ratio tests of Anderson (1963) and their pseudoGaussian robustifications by Tyler (1981, 1983) and Davis (1977), with which their Gaussian versions are shown to coincide, asymptotically, under Gaussian or finite fourthorder moment assumptions, respectively. Such assumptions however restrict the scope to covariancebased principal component analysis. The rankbased tests we are proposing remain valid without such assumptions. Hence, they address a much broader class of problems, where covariance matrices need not exist and principal components are associated with more general scatter matrices. Asymptotic relative efficiencies moreover show that those rankbased tests are quite powerful; when based on van der Waerden or normal scores, they even uniformly dominate the pseudoGaussian versions
A Chernoff–Savage result for shape. On the nonadmissibility of pseudoGaussian methods
 J. Multivariate Anal
, 2006
"... Chernoff and Savage (1958) established that, in the context of univariate location models, Gaussianscore rankbased procedures uniformly dominate—in terms of Pitman asymptotic relative efficiencies—their pseudoGaussian parametric counterparts. This result, which had quite an impact on the success ..."
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Cited by 11 (9 self)
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Chernoff and Savage (1958) established that, in the context of univariate location models, Gaussianscore rankbased procedures uniformly dominate—in terms of Pitman asymptotic relative efficiencies—their pseudoGaussian parametric counterparts. This result, which had quite an impact on the success and subsequent development of rankbased inference, has been extended to many location problems, including problems involving multivariate and/or dependent observations. In this paper, we show that this uniform dominance also holds in problems for which the parameter of interest is the shape of an elliptical distribution. The Pitman nonadmissibility of the pseudoGaussian maximum likelihood estimator for shape and that of the pseudoGaussian likehood ratio test of sphericity follow.
A general method for constructing pseudoGaussian tests
 J. Japan Statist. Soc
, 2008
"... A general method for constructing pseudoGaussian tests—reducing to traditional Gaussian tests under Gaussian densities but remaining valid under nonGaussian ones—is proposed. This method provides a solution to several open problems in classical multivariate analysis. One of them is the test of t ..."
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Cited by 9 (3 self)
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A general method for constructing pseudoGaussian tests—reducing to traditional Gaussian tests under Gaussian densities but remaining valid under nonGaussian ones—is proposed. This method provides a solution to several open problems in classical multivariate analysis. One of them is the test of the homogeneity of covariance matrices, an assumption that plays a crucial role in multivariate analysis of variance, under elliptical, and possibly heterokurtic densities with finite fourthorder moments. Key words and phrases: Elliptical symmetry, homogeneity of covariances, local asymptotic normality, multivariate analysis of variance, pseudoGaussian tests. 1.
Restricted Canonical Correlations
, 1993
"... Given a pdimensional random variable yell and a qdimensional random variabley(2), the first canonical correlation leads to finding Q * E Rp and (3 * E R q which maximizes the correlation between Q'y(l) and (3'y(2). However, in many practical situations (e.g. educational testing problems ..."
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Cited by 9 (0 self)
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Given a pdimensional random variable yell and a qdimensional random variabley(2), the first canonical correlation leads to finding Q * E Rp and (3 * E R q which maximizes the correlation between Q'y(l) and (3'y(2). However, in many practical situations (e.g. educational testing problems, neural networks), some natural restrictions on the coefficients Q and (3 may arise which should be incorporated in the maximization procedure. The maximum correlation subject to these constraints is referred to as restricted canonical correlations. In this work the solution is obtained to the problem under the nonnegativity restriction on Q and (3. The analysis is extended"to more general form of inequality constraints, and also when the restrictions are present only on some of the coefficients. Also discussed are restricted versions of some other multivariate methods. This includes principal component analysis and different modifications of canonical correlation analysis. Some properties of restricted canonical correlations including its bounds have been studied. Since the restricted canonical correlation depends only on the covariance matrix, its sample version can naturally be obtained from sample covariance matrix. The asymptotic normality of this sample restricted canonical correlation is proved under reasonably mild conditions. The study of resampling methods becomes necessary because the asymptotic variance involves usually unknown fourth order moments of the population. The effectiveness of the jackknife method is shown in this case as well as for the usual canonical correlations. Bootstrapping also works out in both these cases. These theoretical results have been supplemented by simulation studies which compare the performances of the two resampling methods in the case of finite samples.
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
On normal theory and associated test statistics in covariance structure analysis under two classes of nonnormal distributions
 Statistica Sinica
, 1999
"... Abstract: Several test statistics for covariance structure models derived from the normal theory likelihood ratio are studied. These statistics are robust to certain violations of the multivariate normality assumption underlying the classical method. In order to explicitly model the behavior of the ..."
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Cited by 5 (2 self)
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Abstract: Several test statistics for covariance structure models derived from the normal theory likelihood ratio are studied. These statistics are robust to certain violations of the multivariate normality assumption underlying the classical method. In order to explicitly model the behavior of these statistics, two new classes of nonnormal distributions are defined and their fourthorder moment matrices are obtained. These nonnormal distributions can be used as alternatives to elliptical symmetric distributions in the study of the robustness of a multivariate statistical method. Conditions for the validity of the statistics under the two classes of nonnormal distributions are given. Some commonly used models are considered as examples to verify our conditions under each class of nonnormal distributions. It is shown that these statistics are valid under much wider classes of distributions than previously assumed. The theory also provides an explanation for previously reported MonteCarlo results on some of the statistics.
Copula Structure Analysis
, 2009
"... In this paper we extend the standard approach of correlation structure analysis for dimension reduction of highdimensional statistical data. The classical assumption of a linear model for the distribution of a random vector is replaced by the weaker assumption of a model for the copula. For elliptic ..."
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Cited by 5 (0 self)
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In this paper we extend the standard approach of correlation structure analysis for dimension reduction of highdimensional statistical data. The classical assumption of a linear model for the distribution of a random vector is replaced by the weaker assumption of a model for the copula. For elliptical copulae a correlationlike structure remains, but different margins and nonexistence of moments are possible. After introducing the new concept and deriving some theoretical results we observe in a simulation study the performance of the estimators: the theoretical asymptotic behavior of the statistics can be observed even for small sample sizes. Finally, we show our method at work for a financial data set and explain differences between our copula based approach and the classical approach. Our new method yields a considerable dimension reduction also in nonlinear models.