The Asymptotic Properties of the Extreme Eigenvectors of High-dimensional Generalized Spiked Covariance Model
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910446537146368 |
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| author | Pu, Zhangni Zhang, Xiaozhuo Hu, Jiang Bai, Zhidong |
| author_facet | Pu, Zhangni Zhang, Xiaozhuo Hu, Jiang Bai, Zhidong |
| contents | In this paper, we investigate the asymptotic behaviors of the extreme eigenvectors in a general spiked covariance matrix, where the dimension and sample size increase proportionally. We eliminate the restrictive assumption of the block diagonal structure in the population covariance matrix. Moreover, there is no requirement for the spiked eigenvalues and the 4th moment to be bounded. Specifically, we apply random matrix theory to derive the convergence and limiting distributions of certain projections of the extreme eigenvectors in a large sample covariance matrix within a generalized spiked population model. Furthermore, our techniques are robust and effective, even when spiked eigenvalues differ significantly in magnitude from nonspiked ones. Finally, we propose a powerful statistic for hypothesis testing for the eigenspaces of covariance matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08524 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Asymptotic Properties of the Extreme Eigenvectors of High-dimensional Generalized Spiked Covariance Model Pu, Zhangni Zhang, Xiaozhuo Hu, Jiang Bai, Zhidong Statistics Theory In this paper, we investigate the asymptotic behaviors of the extreme eigenvectors in a general spiked covariance matrix, where the dimension and sample size increase proportionally. We eliminate the restrictive assumption of the block diagonal structure in the population covariance matrix. Moreover, there is no requirement for the spiked eigenvalues and the 4th moment to be bounded. Specifically, we apply random matrix theory to derive the convergence and limiting distributions of certain projections of the extreme eigenvectors in a large sample covariance matrix within a generalized spiked population model. Furthermore, our techniques are robust and effective, even when spiked eigenvalues differ significantly in magnitude from nonspiked ones. Finally, we propose a powerful statistic for hypothesis testing for the eigenspaces of covariance matrices. |
| title | The Asymptotic Properties of the Extreme Eigenvectors of High-dimensional Generalized Spiked Covariance Model |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2405.08524 |