{
  "abstract": "In nonparametric density estimation without privacy constraints, it is known that a confidence set whose coverage holds uniformly over a `rough' class of regularity $r\\geq0$ can simultaneously attain, in $L^2$-diameter, the minimax rate over a smaller, smoother class of regularity $s>r$ if and only if $s \\le 2r$. We study such adaptive honest confidence sets under non-interactive $α$-local differential privacy (LDP) constraints. Our main result is that, for bounded $α$, such `smooth-rate adaptation' is now possible if and only if $s \\le 4r/3$. In this smaller range we construct an honest private confidence set whose diameter is of the private minimax order on each of the two classes. As in the non-private case, the threshold arises because an honest set cannot, at a smooth density, be narrower than the separation rate arising in a related testing problem. Under LDP we prove that this private testing rate is of order $(nα^2)^{-2r/(4r+3)}$, which is at most the private estimation rate $(nα^2)^{-s/(2s+2)}$ over the smooth class exactly when $s \\le 4r/3$. For $s > 4r/3$, however, we show that smooth-rate adaptation becomes possible once one discards from the rough class the densities that lie outside the smooth class but within an $L^2$-radius of the order of the private testing rate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15982",
  "author_total": 2,
  "authors": [
    {
      "name": "Aditya Khan"
    },
    {
      "name": "Thibault Randrianarisoa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86ycck8eq7b17.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15982",
  "fulltext_url": null,
  "id": "86ycck8eq7b17",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15982",
  "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.TH"
    ],
    [
      "openalex",
      "1702|Privacy-Preserving Technologies in Data"
    ]
  ],
  "openalex": "W7221815882",
  "pdf_url": "https://arxiv.org/pdf/2610.15982",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86ycck8eq7b17.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15982",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15982"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Adaptive Confidence Sets for Nonparametric Density Estimation under Local Differential Privacy",
  "title_text": "Adaptive Confidence Sets for Nonparametric Density Estimation under Local Differential Privacy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/86ycck8eq7b17",
  "venue": "arXiv",
  "versions": []
}