{
  "abstract": "We determine sharp minimax rates for distribution estimation under pure local differential privacy at critical query smoothness. For the integral probability metric generated by the unit ball of $B^d_{2,q}$, $1\\le q\\le\\infty$, over fixed periodic Sobolev density balls with positive smoothness and interior slack at the uniform density, the expected risk is of order $N^{-1/2}(\\log N)^{3/2-1/q}$, where $N=n\\varepsilon^2$; expected squared risk has the squared order. The fine index governs aggregation across resolutions that share one private information budget. A single noninteractive release with a measurable proper density output attains these rates, and matching lower bounds allow finite public blackboard interaction and repeated participation under each user's total privacy budget. A known $L^2$ radius permits the same rates without using density smoothness, with output in the $L^2$ density ball. Separately, for positive smooth densities on $[0,1]$, integrated squared CDF risk has minimax order $(\\log N)^2/N$, with sharp quantile, Wasserstein, and transport map consequences. The rates hold uniformly for $0<\\varepsilon\\le1$ as $N\\to\\infty$, with class parameters fixed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01289",
  "author_total": 3,
  "authors": [
    {
      "name": "Yi Liu"
    },
    {
      "name": "Xiaotian Chang"
    },
    {
      "name": "Qirui Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dq0y2tm4z6qef.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01289",
  "fulltext_url": null,
  "id": "dq0y2tm4z6qef",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01289",
  "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.TH"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01289",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dq0y2tm4z6qef.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01289",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01289"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Sharp Critical Rates for Locally Private Distribution Estimation",
  "title_text": "Sharp Critical Rates for Locally Private Distribution Estimation",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:35Z",
  "url": "https://research.draconitic.com/paper/dq0y2tm4z6qef",
  "venue": "arXiv",
  "versions": []
}