{"ID":2841580,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.11054","arxiv_id":"2511.11054","title":"Tuning free Catoni type joint robust estimation","abstract":"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.","short_abstract":"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 cano...","url_abs":"https://arxiv.org/abs/2511.11054","url_pdf":"https://arxiv.org/pdf/2511.11054v2","authors":"[\"Xiang Li\",\"Jun S. Liu\",\"Qiang Sun\",\"Lihu Xu\"]","published":"2025-11-14T08:10:35Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}