Asymptotic distribution of spiked eigenvalues in the large signal-plus-noise models
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916101101715456 |
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| author | Lin, Zeqin Pan, Guangming Zhao, Peng Zhou, Jia |
| author_facet | Lin, Zeqin Pan, Guangming Zhao, Peng Zhou, Jia |
| contents | Consider large signal-plus-noise data matrices of the form $S + Σ^{1/2} X$, where $S$ is a low-rank deterministic signal matrix and the noise covariance matrix $Σ$ can be anisotropic. We establish the asymptotic joint distribution of its spiked singular values when the dimensionality and sample size are comparably large and the signals are supercritical under general assumptions concerning the structure of $(S, Σ)$ and the distribution of the random noise $X$. It turns out that the asymptotic distributions exhibit nonuniversality in the sense of dependence on the distributions of the entries of $X$, which contrasts with what has previously been established for the spiked sample eigenvalues in the context of spiked population models. Such a result yields the asymptotic distribution of the sample spiked eigenvalues associated with mixture models. We also explore the application of these findings in detecting mean heterogeneity of data matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_11672 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic distribution of spiked eigenvalues in the large signal-plus-noise models Lin, Zeqin Pan, Guangming Zhao, Peng Zhou, Jia Statistics Theory 60F05, 60E05 Consider large signal-plus-noise data matrices of the form $S + Σ^{1/2} X$, where $S$ is a low-rank deterministic signal matrix and the noise covariance matrix $Σ$ can be anisotropic. We establish the asymptotic joint distribution of its spiked singular values when the dimensionality and sample size are comparably large and the signals are supercritical under general assumptions concerning the structure of $(S, Σ)$ and the distribution of the random noise $X$. It turns out that the asymptotic distributions exhibit nonuniversality in the sense of dependence on the distributions of the entries of $X$, which contrasts with what has previously been established for the spiked sample eigenvalues in the context of spiked population models. Such a result yields the asymptotic distribution of the sample spiked eigenvalues associated with mixture models. We also explore the application of these findings in detecting mean heterogeneity of data matrices. |
| title | Asymptotic distribution of spiked eigenvalues in the large signal-plus-noise models |
| topic | Statistics Theory 60F05, 60E05 |
| url | https://arxiv.org/abs/2401.11672 |