{
  "abstract": "We study asymmetric simple binary hypothesis testing between $H_0:P_0^{n}$ and $H_1:P_1^{n}$, based on $n$ independent and identically distributed observations. Leveraging a variational representation of Rényi divergence of order $α$, we derive our main result: a finite-sample converse with $α>1$. The bound uses both directions of the divergence $D_α(P_1\\|P_0)$ and $D_α(P_0\\|P_1)$, tensorises under product measures, and contains familiar data-processing converses as boundary cases. For comparison, we apply the same variational approach to general $f$-divergences and specialise it to total variation, $E_γ$, Hellinger, and Kullback Leibler divergences, thereby recovering familiar converses within a unified framework. Together with an achievability bound involving Rényi divergence with $α\\in (0,1)$, the main converse recovers the phase transition of the optimal Type II error under the exponentially decaying Type I error constraint $\\varepsilon_n=e^{-nr}$. Under regularity conditions, the optimal Type II error vanishes exponentially when $r<D(P_1\\|P_0)$ and converges exponentially fast to one when $r>D(P_1\\|P_0)$. We also derive sample-complexity bounds and extend both the converse and achievability analyses to locally differentially private observations, quantifying the cost of privacy and recovering the non-private achievability bound as the privacy constraint vanishes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27617",
  "author_total": 3,
  "authors": [
    {
      "name": "Roberto Bruno"
    },
    {
      "name": "Adrien Vandenbroucque"
    },
    {
      "name": "Amedeo Roberto Esposito"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1mc5n2wbzx0q5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27617",
  "fulltext_url": null,
  "id": "1mc5n2wbzx0q5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27617",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1705|Distributed Sensor Networks and Detection Algorithms"
    ]
  ],
  "openalex": "W7214155468",
  "pdf_url": "https://arxiv.org/pdf/2609.27617",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1mc5n2wbzx0q5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27617",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27617"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Finite-Sample Binary Hypothesis Testing via Rényi Divergences: Strong Converse and Local Privacy",
  "title_text": "Finite-Sample Binary Hypothesis Testing via Rényi Divergences: Strong Converse and Local Privacy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:06Z",
  "url": "https://research.draconitic.com/paper/1mc5n2wbzx0q5",
  "venue": "arXiv",
  "versions": []
}