On the sphericity test with large-dimensional observations

Wang, Qinwen; Yao, Jianfeng

doi:10.1214/13-ejs842

Cited by 66 publications

(52 citation statements)

References 34 publications

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“…This test has been extended to the high dimensional framework in a series of recent works such as Ledoit and Wolf (), Birke and Dette (), Wang and Yao () and Tian et al . ().…”

Section: Testing the Sphericity Of A High Dimensional Mixturementioning

confidence: 99%

“…In this section, using the results that were developed in Section 2, we theoretically investigate the reliability of John's test for the sphericity of a covariance matrix (John, 1972) and its high dimensional corrected version (Wang and Yao, 2013) when the underlying distribution is a high dimensional mixture. Our findings show that neither John's test nor its corrected version is thus valid any longer.…”

Section: Testing the Sphericity Of A High Dimensional Mixturementioning

confidence: 99%

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On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

Yao

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

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By studying the family of p-dimensional scale mixtures, this paper shows for the first time a non trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated Marčenko-Pastur law. A different and new limit is found and characterized. The reasons of failure of the Marčenko-Pastur limit in this situation are found to be a strong dependence between the p-coordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analize the traditional John's type test we establish a novel and general CLT for linear statistics of eigenvalues of the sample covariance matrix. It is shown that the John's test and its recent high-dimensional extensions both fail for high-dimensional mixtures, precisely due to the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used ICL and BIC criteria in detecting non spherical component covariance matrices of a high-dimensional mixture.

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“…This test has been extended to the high dimensional framework in a series of recent works such as Ledoit and Wolf (), Birke and Dette (), Wang and Yao () and Tian et al . ().…”

Section: Testing the Sphericity Of A High Dimensional Mixturementioning

confidence: 99%

Section: Testing the Sphericity Of A High Dimensional Mixturementioning

confidence: 99%

On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

Yao

2017

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…By contrast, LRT is only valid in the case of small n and large p, that is, n should be no more than 40. As to this point, there are some literatures that extend LRT to the large-dimensional case (see Wang and Yao 2013).…”

Section: Testing Independencementioning

confidence: 99%

Testing Independence Among a Large Number of High-Dimensional Random Vectors

Pan

Gao

Yang

2014

Journal of the American Statistical Association

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Capturing dependence among a large number of high-dimensional random vectors is a very important and challenging problem. By arranging n random vectors of length p in the form of a matrix, we develop a linear spectral statistic of the constructed matrix to test whether the n random vectors are independent or not. Specifically, the proposed statistic can also be applied to n random vectors, each of whose elements can be written as either a linear stationary process or a linear combination of independent random variables. The asymptotic distribution of the proposed test statistic is established for the case of 0 < lim n→∞ p n < ∞ as n → ∞. To avoid estimating the spectrum of each random vector, a modified test statistic, which is based on splitting the original n vectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series, is constructed. The facts that the limiting distribution is normal and there is no need to know the inner structure of each investigated random vector result in simple implementation of the constructed test statistic. Simulation results demonstrate that the proposed test is powerful against several commonly used dependence structures. An empirical application to detecting dependence of the closed prices from several stocks in the S&P500 also illustrates the applicability and effectiveness of our provided test. Supplementary materials for this article are available online.

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“…For scenarios with multiple spikes, there are relatively few existing results concerning LSS, and the results which are available focus primarily on Johnstone's spiked model. These include [WSY13], which derived a CLT for general LSS, expressing the limiting mean and variance in terms of contour integrals, as well as [Ona14,PY13,WY13,OMH14], which considered specific linear statistics (i.e., for specific applications). For alternative spiked models, such as the non-central Wishart and F scenarios, results concerning LSS are currently absent, beyond the single-spike scenario considered in [PMC14].…”

Section: Introductionmentioning

confidence: 99%

Hypergeometric functions of matrix arguments and linear statistics of multi-spiked Hermitian matrix models

Passemier

McKay

Chen

2015

Journal of Multivariate Analysis

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Abstract. This paper derives central limit theorems (CLTs) for general linear spectral statistics (LSS) of three important multi-spiked Hermitian random matrix ensembles. The first is the most common spiked scenario, proposed by Johnstone, which is a central Wishart ensemble with fixed-rank perturbation of the identity matrix, the second is a non-central Wishart ensemble with fixed-rank noncentrality parameter, and the third is a similarly defined non-central F ensemble. These CLT results generalize our recent work [PMC14] to account for multiple spikes, which is the most common scenario met in practice. The generalization is non-trivial, as it now requires dealing with hypergeometric functions of matrix arguments. To facilitate our analysis, for a broad class of such functions, we first generalize a recent result of Onatski [Ona14] to present new contour integral representations, which are particularly suitable for computing large-dimensional properties of spiked matrix ensembles. Armed with such representations, our CLT formulas are derived for each of the three spiked models of interest by employing the Coulomb fluid method from random matrix theory along with saddlepoint techniques. We find that for each matrix model, and for general LSS, the individual spikes contribute additively to yield a O(1) correction term to the asymptotic mean of the linear statistic, which we specify explicitly, whilst having no effect on the leading order terms of the mean or variance.

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On the sphericity test with large-dimensional observations

Cited by 66 publications

References 34 publications

On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

On Structure Testing for Component Covariance Matrices of a High Dimensional Mixture

Testing Independence Among a Large Number of High-Dimensional Random Vectors

Hypergeometric functions of matrix arguments and linear statistics of multi-spiked Hermitian matrix models

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