Tuning free Catoni type joint robust estimation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917501512712192 |
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| author | Li, Xiang Liu, Jun S. Sun, Qiang Xu, Lihu |
| author_facet | Li, Xiang Liu, Jun S. Sun, Qiang Xu, Lihu |
| contents | This paper develops a Catoni-type joint (tuning-free) estimation framework for parametric models with heavy-tailed noise, in which the target parameter and the unknown noise variance are estimated simultaneously through a system of two coupled Catoni-type estimating equations. We instantiate the framework in three canonical settings: mean estimation, linear regression, and $\ell_{2}$-penalized regression.
Theoretically, we establish non-asymptotic, sub-Gaussian-type deviation bounds that hold jointly for the target parameter and the variance estimator, under only a finite $2β$-th moment assumption with $β\in (1,2]$. The resulting rates match -- up to absolute constants -- those of oracle procedures that know the variance in advance, thereby attaining optimality in the heavy-tailed regime.
Methodologically, because the coupled equations are intrinsically non-convex and non-linear, classical convex M-estimation arguments are inapplicable. We develop a new analytical toolkit based on the Poincare--Miranda theorem. The resulting proof strategy is of independent methodological interest, and we expect it to be applicable to a broad class of other statistical problems in which several parameters of heterogeneous nature must be estimated jointly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11054 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tuning free Catoni type joint robust estimation Li, Xiang Liu, Jun S. Sun, Qiang Xu, Lihu Statistics Theory Primary 62G35, 62F35, Secondary 62J07, 62H12 This paper develops a Catoni-type joint (tuning-free) estimation framework for parametric models with heavy-tailed noise, in which the target parameter and the unknown noise variance are estimated simultaneously through a system of two coupled Catoni-type estimating equations. We instantiate the framework in three canonical settings: mean estimation, linear regression, and $\ell_{2}$-penalized regression. Theoretically, we establish non-asymptotic, sub-Gaussian-type deviation bounds that hold jointly for the target parameter and the variance estimator, under only a finite $2β$-th moment assumption with $β\in (1,2]$. The resulting rates match -- up to absolute constants -- those of oracle procedures that know the variance in advance, thereby attaining optimality in the heavy-tailed regime. Methodologically, because the coupled equations are intrinsically non-convex and non-linear, classical convex M-estimation arguments are inapplicable. We develop a new analytical toolkit based on the Poincare--Miranda theorem. The resulting proof strategy is of independent methodological interest, and we expect it to be applicable to a broad class of other statistical problems in which several parameters of heterogeneous nature must be estimated jointly. |
| title | Tuning free Catoni type joint robust estimation |
| topic | Statistics Theory Primary 62G35, 62F35, Secondary 62J07, 62H12 |
| url | https://arxiv.org/abs/2511.11054 |