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Some tests concerning the covariance matrix in HighDimensional data
 J. Japan. Statist. Soc
, 2005
"... In this paper, tests are developed for testing certain hypotheses on the covariance matrix Σ, when the sample size N = n+1 is smaller than the dimension p of the data. Under the condition that (trΣi/p) exists and> 0, as p→∞, i = 1,..., 8, tests are developed for testing the hypotheses that the c ..."
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Cited by 29 (8 self)
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In this paper, tests are developed for testing certain hypotheses on the covariance matrix Σ, when the sample size N = n+1 is smaller than the dimension p of the data. Under the condition that (trΣi/p) exists and> 0, as p→∞, i = 1,..., 8, tests are developed for testing the hypotheses that the covariance matrix in a normally distributed data is an identity matrix, a constant time the identity matrix (spherecity), and is a diagonal matrix. The asymptotic null and nonnull distributions of these test statistics are given. Key words and phrases: Asymptotic distributions, multivariate normal, null and nonnull distributions, sample size smaller than the dimension. 1.
Optimal hypothesis testing for high dimensional covaraiance matrices
 Bernoulli
, 2013
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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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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
Detecting Positive Correlations in a Multivariate Sample
, 2012
"... We consider the problem of testing whether a correlation matrix of a multivariate normal population is the identity matrix. We focus on sparse classes of alternatives where only a few entries are nonzero and, in fact, positive. We derive a general lower bound applicable to various classes and study ..."
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Cited by 7 (1 self)
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We consider the problem of testing whether a correlation matrix of a multivariate normal population is the identity matrix. We focus on sparse classes of alternatives where only a few entries are nonzero and, in fact, positive. We derive a general lower bound applicable to various classes and study the performance of some nearoptimal tests. We pay special attention to computational feasibility and construct nearoptimal tests that can be computed efficiently. Finally, we apply our results to prove new lower bounds for the clique number of highdimensional random geometric graphs.
Maximum Covariance Difference Test for Equality of Two Covariance Matrices
 In Algebraic Methods in Statistics and Probability
"... We propose a test ofeqOIIk y of two covariance matrices based on the maximum standardized di#erence of scalar covariances of two sample covariance matrices. We derive the tail probability of the asymptotic null distribution of the test statistic by the tube method. However the usual formal tube form ..."
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Cited by 1 (1 self)
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We propose a test ofeqOIIk y of two covariance matrices based on the maximum standardized di#erence of scalar covariances of two sample covariance matrices. We derive the tail probability of the asymptotic null distribution of the test statistic by the tube method. However the usual formal tube formula has to be suitably modified, because in this case the index set, around which the tube is formed, has zero critical radius. 1.
On Some PatternReduction Matrices Which Appear in Statistics
"... We review and extend some recent results concerning the structure of patternreduction matrices, which effect the reduction of the vet of a patterned matrix to the vector consisting only of the functionally independent elements of the matrix. The results are applied to the calculation of certain Jac ..."
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We review and extend some recent results concerning the structure of patternreduction matrices, which effect the reduction of the vet of a patterned matrix to the vector consisting only of the functionally independent elements of the matrix. The results are applied to the calculation of certain Jacobians, and to the construction of ellipsoidal confidence regions for covariance matrices, on the basis of maximum likelihood or robust Mestimators. 1.
Acknowledgements
"... In preparing this thesis I am indebted to many people. First of all, I wish to express my sincere gratitude to Professor Chihiro Hirotsu, my thesis supervisor, for his guidance and encouragement. I would like to express my thanks to Professor Kei Takeuchi, Professor Shunichi Amari, Professor Yasuo ..."
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In preparing this thesis I am indebted to many people. First of all, I wish to express my sincere gratitude to Professor Chihiro Hirotsu, my thesis supervisor, for his guidance and encouragement. I would like to express my thanks to Professor Kei Takeuchi, Professor Shunichi Amari, Professor Yasuo Ohashi, Professor Akimichi Takemura and Dr.