{
  "abstract": "The sub-Gaussian parameter (also called the variance proxy) of a mean-zero random variable $X$ is defined as $ξ^2_\\star = \\sup_{λ\\in \\mathbb{R}} L(λ)$ where $L(λ) = \\frac{2}{λ^2} \\log \\mathbb{E} e^{λX}$ is a weighted cumulant generating function. We study the estimation of $ξ^2_\\star$ and prove that the minimax risk is governed by a non-increasing function $δ_P(C) = \\sup_{|λ| \\geq C} L(λ) - \\sup_{|λ| \\leq C} L(λ)$ which captures the influence of the tail behavior of the distribution $P$. Over the class of distributions with $δ_P \\leq r$ for a non-increasing function $r$, the minimax risk is, up to a multiplicative constant, lower bounded by $r(\\sqrt{\\log n}) + n^{-1/2}$ and upper bounded by $r((\\log n)^{1/2-\\varepsilon}) + n^{-1/2 + \\varepsilon}$ for any $\\varepsilon > 0$. Our estimator for the upper bound is based on constrained maximization of the empirical analogue of $L$. In addition to being almost minimax optimal and adaptive, we further prove that the estimator is asymptotic normal under suitable conditions and that, if the underlying distribution is not sub-Gaussian, the estimator diverges with a rate determined by the heaviness of the distributional tail.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.06384",
  "author_total": 3,
  "authors": [
    {
      "name": "Jason Liu"
    },
    {
      "name": "Min Xu"
    },
    {
      "name": "Jinchuan Xing"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3ts25h54nm8q7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.06384",
  "fulltext_url": null,
  "id": "3ts25h54nm8q7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.06384",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Probability and Statistical Research"
    ]
  ],
  "openalex": "W7163687366",
  "pdf_url": "https://arxiv.org/pdf/2606.06384",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3ts25h54nm8q7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2606.06384",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.06384"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Estimation of the sub-Gaussian Parameter",
  "title_text": "Estimation of the sub-Gaussian Parameter",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:03Z",
  "url": "https://research.draconitic.com/paper/3ts25h54nm8q7",
  "venue": "arXiv",
  "versions": []
}