A note on testing the covariance matrix for large dimension

In this paper, we propose corrections to the likelihood ratio test and John's test for sphericity in large-dimensions. New formulas for the limiting parameters in the CLT for linear spectral statistics of sample covariance matrices with general fourth moments are first established. Using these formulas, we derive the asymptotic distribution of the two proposed test statistics under the null. These asymptotics are valid for general population, i.e. not necessarily Gaussian, provided a finite fourth-moment. Extensive Monte-Carlo experiments are conducted to assess the quality of these tests with a comparison to several existing methods from the literature. Moreover, we also obtain their asymptotic power functions under the alternative of a spiked population model as a specific alternative.MSC 2010 subject classifications: Primary 62H15; secondary 62H10.

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“…It is worth noticing that [7] has extended the LW test to such a scheme (p ≫ n) for multivariate normal distribution.…”

Section: Monte Carlo Studymentioning

confidence: 99%

On the sphericity test with large-dimensional observations

Wang¹,

Yao

2013

Electron. J. Statist.

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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).…”

Section: Testing the Identity Of Covariance Matricesmentioning

confidence: 99%

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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“…8. Birke and Dette [5] extend the work of Ledoit and Wolf by considering the same test statistics for the extreme boundaries of concentration, i.e. when p/n → c ∈ [0, ∞].…”

Section: Introductionmentioning

confidence: 98%

Tests for high-dimensional covariance matrices using the theory ofU-statistics

Ahmad

Rosen

2014

Journal of Statistical Computation and Simulation

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Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Under certain mild conditions mainly on the traces of the unknown covariance matrix, and using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p n. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. A real data set is used to illustrate the application of the proposed test statistics.

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A note on testing the covariance matrix for large dimension

Cited by 68 publications

References 9 publications

On the sphericity test with large-dimensional observations

On the sphericity test with large-dimensional observations

Contributions to High–Dimensional Analysis under Kolmogorov Condition

Tests for high-dimensional covariance matrices using the theory ofU-statistics

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