{
  "abstract": "To design systems that are protected against events much rarer than the observational record, extreme-value methods are needed to extrapolate distribution tails. Tail-index estimators such as the Hill estimator are central to this extrapolation, but overestimating the tail exponent can lead to substantial underestimation of rare-event probabilities. Motivated by this, we derive large- and moderate-deviation asymptotics for the Hill estimator and use them to construct estimators whose probability of exceeding the true tail index decays at a controlled exponential rate (the decay rate). In the large-deviations regime, we show that a simple rescaled version of the Hill estimator achieves an optimal balance between bias and decay rate among scale-invariant estimators based on the same top $k$ order statistics. Under a second-order condition, we quantify the effect of the Hill bias, analyze a bias-corrected estimator, and identify sufficient conditions for moderate deviations in the boundary case where the second-order parameter $ρ$ equals zero.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31127",
  "author_total": 3,
  "authors": [
    {
      "name": "Martijn Gösgens"
    },
    {
      "name": "Bart P. G. van Parys"
    },
    {
      "name": "Bert Zwart"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b1bs3jnv3daeq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31127",
  "fulltext_url": null,
  "id": "b1bs3jnv3daeq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31127",
  "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.PR"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1803|Probability and Risk Models"
    ]
  ],
  "openalex": "W7214615056",
  "pdf_url": "https://arxiv.org/pdf/2609.31127",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b1bs3jnv3daeq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31127",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31127"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Large and Moderate Deviations for Conservative Tail-Index Estimation",
  "title_text": "Large and Moderate Deviations for Conservative Tail-Index Estimation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:03Z",
  "url": "https://research.draconitic.com/paper/b1bs3jnv3daeq",
  "venue": "arXiv",
  "versions": []
}