Tests for High-Dimensional Covariance Matrices

Chen, Song Xi; Zhang, Li Xin; Zhong, Ping

doi:10.1198/jasa.2010.tm09560

Cited by 255 publications

(87 citation statements)

References 21 publications

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“…Furthermore, the result of Ledoit and Wolf (2002) has been further developed for lim n,p→∞ n p = c > 0 by Fujikoshi et al (2011) and Birke and Dette (2005). There exists approaches, that origins in Nagao's test statistics, when the normality assumption is omitted, see paper by Chen et al (2010). The publication by Srivastava (2005) also follows the paper of Nagao (1973).…”

Section: Testing the Identity Of Covariance Matricesmentioning

confidence: 86%

Contributions to High–Dimensional Analysis under Kolmogorov Condition

Pielaszkiewicz

2015

Linköping Studies in Science and Technology. Dissertations

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This thesis is about high-dimensional problems considered under the so-called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p > n, assuming that the ratio p n converges when the number of parameters and the sample size increase.We focus on the eigenvalue distribution of the considered matrices, since it is a well-known information-carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non-commutative space of random matrices of size p × p equipped with the functionalHere, the connections with free probability theory arise. In the relation to that field we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant-moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant-moment relation given through non-crossing partitions of the set.Furthermore, we investigate the normalized E[Tr{W mi }] and derive, using the differentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re-establish moments of the Marčenko-Pastur distribution, and hence the recursive relation for the Catalan numbers.In this thesis we also prove that theWe considerwhere mwhich is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness-of-fit approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high-dimensional data (p > n) and the multivariate data (p ≤ n). Populärvetenskaplig sammanfattningI många tillämpningar förekommer slumpmatriser, det vill säga matriser vars element följer någon stokastisk fördelning. Detär ofta intressant att känna till hur egenvärdena för dessa slumpmässiga matriser uppför sig, det vill säga beräkna fördelningen för egenvärdena, den så kallade spektralfördelningen. Egenvärdenä ar informationsbärande objekt, då de ger information om till exempel stabilitet och inversen av den slumpmässiga matrisen genom det minsta egenvärdet.Telekommunikation och teoretisk fysikär två områden där detär intressant att studera slumpmatriser, matrisernas egenvärden och fördelningen för dessa egenvärden. Speciellt gäller det för stora slumpmatriser där man dåär intresserad av den asymptotiska spektralfördelningen. Ett exempel kan vara en kanalmatris X för ett flerdimensionellt kommunikationssystem, där fördelningen för egenvärdena för matrisen XX * bestämmer kanalkapaciteten och uppnåeligä overföringshastigheter.Spektralfördelningen har studerats ingående inom teorin för slumpmässiga matriser. I denna avhandling kombinerar vi resulta...

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Section: Testing the Identity Of Covariance Matricesmentioning

confidence: 86%

Contributions to High–Dimensional Analysis under Kolmogorov Condition

Pielaszkiewicz

2015

Linköping Studies in Science and Technology. Dissertations

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“…Chen et al [15] extended the work to the case without normality assumptions. Now, the target functions (3.1) can be rewritten as…”

Section: Some Related Work On the Tests Of High-dimensional Locationsmentioning

confidence: 99%

A review of 20 years of naive tests of significance for high-dimensional mean vectors and covariance matrices

Bai

2016

Sci. China Math.

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In this paper, we introduce the so-called naive tests and give a brief review of the new developments. Naive testing methods are easy to understand and perform robustly, especially when the dimension is large. In this paper, we focus mainly on reviewing some naive testing methods for the mean vectors and covariance matrices of high-dimensional populations, and we believe that this naive testing approach can be used widely in many other testing problems.

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“…This builds on the paper by Chen et al (2010) who use U -statistics to test for sphericity of the variance-covariance matrix in statistics. The null of sphericity means that the variance-covariance matrix is proportional to the identity matrix.…”

Section: Introductionmentioning

confidence: 99%

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AbstractBuilding upon the work of Chen et al (2010), this paper proposes a test for sphericity of the variance-covariance matrix in a fixed effects panel data regression model without the normality assumption on the disturbances.

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confidence: 99%