{
  "abstract": "Location estimation exhibits markedly different finite-sample behavior across noise distributions: regular families typically yield root-\\(n\\) rates, whereas compactly supported laws may admit faster, boundary-driven rates. We question whether a single estimator, without knowledge of the density's shape, can adapt to the instance-wise optimal estimation rate, as an oracle that knows the underlying location family can. For a known location family with symmetric log-concave noise density \\(f\\), the optimal location estimation error with sample size \\(n\\) under failure probability \\(δ\\) is known to be Le Cam's two-point rate: \\[ \\sup\\left\\{r>0:\\mathsf{H}^2\\left(f_0, f_{2r}\\right)\\lesssim \\frac{\\log(1/δ)}{n}\\right\\}. \\] When the location family is unknown, we propose a shape-agnostic estimator that attains this oracle benchmark simultaneously over all symmetric unimodal densities with non-decreasing hazard rates, a class strictly broader than symmetric log-concave distributions. We establish that the Hellinger-driven two-point rate can be characterized solely by a multiscale function of dyadic quantile gaps. This new structural connection between Hellinger divergence and quantile geometry motivates a simple estimation procedure that aggregates sample mid-summaries with carefully designed data-dependent weights. The resulting estimator is finite-sample instance-optimal and runs in only \\(O(\\log(n))\\) time on sorted samples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20749",
  "author_total": 2,
  "authors": [
    {
      "name": "Qiaosen Wang"
    },
    {
      "name": "Chao Gao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bja6cefmmmwtn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20749",
  "fulltext_url": null,
  "id": "bja6cefmmmwtn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20749",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.ME"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "2613|Statistical Methods and Inference"
    ]
  ],
  "openalex": "W7213607545",
  "pdf_url": "https://arxiv.org/pdf/2609.20749",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bja6cefmmmwtn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20749",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20749"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries",
  "title_text": "Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:29Z",
  "url": "https://research.draconitic.com/paper/bja6cefmmmwtn",
  "venue": "arXiv",
  "versions": []
}