A robust test for sphericity of high-dimensional covariance matrices

Tian, Xintao; Lu, Yuting; Li, Weiming

doi:10.1016/j.jmva.2015.07.010

Cited by 16 publications

(23 citation statements)

References 17 publications

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“…(), Li and Yao (), Tian et al . () and the references therein. In our context, the j th moment statistic can be expressed as

{\overset{β}{true}}_{n j} = \int x^{j} d F_{n} false(x false), j \in N,

and its limit is related to the following four quantities:

\begin{matrix} {italicβ}_{n j} = \int x^{j} d F^{c_{n}, G_{n}} false(x false), \\ {italicβ}_{j} = \int x^{j} d F^{c_{n}, G} false(x false), \\ {italicγ}_{n j} = \int t^{j} d G_{n} false(t false), \\ {italicγ}_{j} = \int t^{j} d G false(t false), \end{matrix}

which are the j th moments of corresponding measures.…”

Section: High Dimensional Theory For Eigenvalues Of Bnmentioning

confidence: 99%

“…Among all the LSSs, the moments of sample eigenvalues are those of the most important statistics. They have been well studied in the literature again under the linear transformation model (5); see Pan and Zhou (2008), Bai et al (2010), Li and Yao (2014), Tian et al (2015) and the references therein. In our context, the jth moment statistic can be expressed aŝ…”

Section: Application Of the Central Limit Theorems To Moments Of Sampmentioning

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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“…(), Li and Yao (), Tian et al . () and the references therein. In our context, the j th moment statistic can be expressed as

{\overset{β}{true}}_{n j} = \int x^{j} d F_{n} false(x false), j \in N,

and its limit is related to the following four quantities:

\begin{matrix} {italicβ}_{n j} = \int x^{j} d F^{c_{n}, G_{n}} false(x false), \\ {italicβ}_{j} = \int x^{j} d F^{c_{n}, G} false(x false), \\ {italicγ}_{n j} = \int t^{j} d G_{n} false(t false), \\ {italicγ}_{j} = \int t^{j} d G false(t false), \end{matrix}

which are the j th moments of corresponding measures.…”

Section: High Dimensional Theory For Eigenvalues Of Bnmentioning

confidence: 99%

Section: Application Of the Central Limit Theorems To Moments Of Sampmentioning

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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“…See, for example, Ledoit and Wolf (2002), Wang and Yao (2013), Tian et al (2015) for the linear transform model in (1.2), while Zou et al (2014) and Paindaveine and Verdebout (2016) for the elliptical model in (1.4). In particular, the test statistic in Tian et al (2015) synthesizes the first four moments of sample eigenvalues, by which it gains extra powers for spike-like alternative covariance matrices. However this statistic is not valid for general elliptical populations (Li and Yao, 2017).…”

Section: Assumption (B)mentioning

confidence: 99%

“…Then in Section 3, based on the derived results, we theoretically investigate the problem of sphericity test for covariance matrices. This is done by discussing a John's-type test from Tian et al (2015) for general alternative models and a likelihood ratio test from Onatski et al (2013) for spiked covariances under arbitrary elliptical distributions. For illustration, the John-type test is applied to analyze a dataset of weekly stock returns in Section 4.…”

mentioning

confidence: 99%

“…Due to its concise form and broad applicability, this kind of test is quite favorable for high dimensional situations and has been extensively studied in recent years. See, for example, Ledoit and Wolf (2002), Wang and Yao (2013), Tian et al (2015) for the linear transform model in (1.2), while Zou et al (2014) and Paindaveine and Verdebout (2016) for the elliptical model in (1.4). In particular, the test statistic in Tian et al (2015) synthesizes the first four moments of sample eigenvalues, by which it gains extra powers for spike-like alternative covariance matrices.…”

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confidence: 99%

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High-dimensional covariance matrices in elliptical distributions with application to spherical test

Hu¹,

Li²,

Li³

et al. 2019

Ann. Statist.

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This paper discusses fluctuations of linear spectral statistics of high-dimensional sample covariance matrices when the underlying population follows an elliptical distribution. Such population often possesses high order correlations among their coordinates, which have great impact on the asymptotic behaviors of linear spectral statistics. Taking such kind of dependency into consideration, we establish a new central limit theorem for the linear spectral statistics in this paper for a class of elliptical populations. This general theoretical result has wide applications and, as an example, it is then applied to test the sphericity of elliptical populations.

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High-dimensional sphericity test by extended likelihood ratio

Wang

2021

Metrika

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A robust test for sphericity of high-dimensional covariance matrices

Cited by 16 publications

References 17 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

High-dimensional covariance matrices in elliptical distributions with application to spherical test

High-dimensional sphericity test by extended likelihood ratio

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