Minimax estimation of functionals in sparse vector model with correlated observations
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917339292762112 |
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| author | Wang, Yuhao Yang, Pengkun Tsybakov, Alexandre B. |
| author_facet | Wang, Yuhao Yang, Pengkun Tsybakov, Alexandre B. |
| contents | We consider the observations of an unknown $s$-sparse vector ${\boldsymbol θ}$ corrupted by Gaussian noise with zero mean and unknown covariance matrix ${\boldsymbol Σ}$. We propose minimax optimal methods of estimating the $\ell_2$ norm of ${\boldsymbol θ}$ and testing the hypothesis $H_0: {\boldsymbol θ}=0$ against sparse alternatives when only partial information about ${\boldsymbol Σ}$ is available, such as an upper bound on its Frobenius norm and the values of its diagonal entries to within an unknown scaling factor. We show that the minimax rates of the estimation and testing are leveraged not by the dimension of the problem but by the value of the Frobenius norm of ${\boldsymbol Σ}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14778 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimax estimation of functionals in sparse vector model with correlated observations Wang, Yuhao Yang, Pengkun Tsybakov, Alexandre B. Statistics Theory We consider the observations of an unknown $s$-sparse vector ${\boldsymbol θ}$ corrupted by Gaussian noise with zero mean and unknown covariance matrix ${\boldsymbol Σ}$. We propose minimax optimal methods of estimating the $\ell_2$ norm of ${\boldsymbol θ}$ and testing the hypothesis $H_0: {\boldsymbol θ}=0$ against sparse alternatives when only partial information about ${\boldsymbol Σ}$ is available, such as an upper bound on its Frobenius norm and the values of its diagonal entries to within an unknown scaling factor. We show that the minimax rates of the estimation and testing are leveraged not by the dimension of the problem but by the value of the Frobenius norm of ${\boldsymbol Σ}$. |
| title | Minimax estimation of functionals in sparse vector model with correlated observations |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2407.14778 |