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Optimal detection of sparse principal components in high dimension
, 2013
"... We perform a finite sample analysis of the detection levels for sparse principal components of a highdimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NPcomplete in general, and we describe a computationally ..."
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Cited by 38 (4 self)
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We perform a finite sample analysis of the detection levels for sparse principal components of a highdimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NPcomplete in general, and we describe a computationally efficient alternative test using convex relaxations. Our relaxation is also proved to detect sparse principal components at near optimal detection levels, and it performs well on simulated datasets. Moreover, using polynomial time reductions from theoretical computer science, we bring significant evidence that our results cannot be improved, thus revealing an inherent trade off between statistical and computational performance.
Central Limit Theorems for Classical Likelihood Ratio Tests for HighDimensional Normal Distributions
"... For random samples of size n obtained from pvariate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the highdimensional setting. These test statistics have been extensively studied in multivariate analysis and their limiting d ..."
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Cited by 11 (4 self)
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For random samples of size n obtained from pvariate normal distributions, we consider the classical likelihood ratio tests (LRT) for their means and covariance matrices in the highdimensional setting. These test statistics have been extensively studied in multivariate analysis and their limiting distributions under the null hypothesis were proved to be chisquare distributions as n goes to infinity and p remains fixed. In this paper, we consider the highdimensional case where both p and n go to infinity with p/n → y ∈ (0, 1]. We prove that the likelihood ratio test statistics under this assumption will converge in distribution to normal distributions with explicit means and variances. We perform the simulation study to show that the likelihood ratio tests using our central limit theorems outperform those using the traditional chisquare approximations for analyzing highdimensional data.
Optimal hypothesis testing for high dimensional covaraiance matrices
 Bernoulli
, 2013
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Supplement to “Universal asymptotics for highdimensional sign tests
, 2013
"... In a smalln largep hypothesis testing framework, most procedures in the literature require quite stringent distributional assumptions, and restrict to a specific scheme of (n, p)asymptotics. More precisely, multinormality is almost always assumed, and it is imposed, typically, that p/n → c, fo ..."
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Cited by 4 (3 self)
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In a smalln largep hypothesis testing framework, most procedures in the literature require quite stringent distributional assumptions, and restrict to a specific scheme of (n, p)asymptotics. More precisely, multinormality is almost always assumed, and it is imposed, typically, that p/n → c, for some c in some given convex set C ⊂ (0,∞). Such restrictions clearly jeopardize practical relevance of these procedures. In this paper, we consider several classical testing problems in multivariate analysis, directional statistics, and multivariate time series: the problem of testing uniformity on the unit sphere, the spherical location problem, the problem of testing that a process is white noise versus serial dependence, the problem of testing for multivariate independence, and the problem of testing for sphericity. In each case, we show that the natural sign tests enjoy nonparametric validity and are distributionfree in a “universal ” (n, p)asymptotics framework, where p may go to infinity in an arbitrary way as n does. Simulations confirm our asymptotic results. 1. Introduction. There
Three problems related to the eigenvalues of complex noncentral Wishart matrices with rank1 mean, submitted to
 SIAM Journal on Matrix Analysis and Applications
, 2013
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Hypergeometric Functions of Matrix Arguments and Linear Statistics of MultiSpiked Hermitian Matrix Models
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HighDimensional Tests for Spherical Location and Spiked Covariance
"... Highdimensional tests for spherical location and spiked covariance ..."
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Highdimensional tests for spherical location and spiked covariance
On the largest Lyapunov exponent for products of Gaussian random matrices
, 1302
"... The paper provides a new formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternionvalued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large. In addition, the ..."
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The paper provides a new formula for the largest Lyapunov exponent of Gaussian matrices, which is valid in the real, complex and quaternionvalued cases. This formula is applied to derive asymptotic expressions for the largest Lyapunov exponent when the size of the matrix is large. In addition, the paper gives new exact formulas for the largest Lyapunov exponents of 3by3 real and dbyd quaternion Gaussian matrices. 1.
Journal of Multivariate Analysis 130 (2014) 194–207 Contents lists available at ScienceDirect Journal of Multivariate Analysis
"... journal homepage: www.elsevier.com/locate/jmva A note on the CLT of the LSS for sample covariance matrix from a spiked population model ..."
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journal homepage: www.elsevier.com/locate/jmva A note on the CLT of the LSS for sample covariance matrix from a spiked population model