{
  "abstract": "We analyze the problem of estimating univariate Gaussian mixture densities using the nonparametric maximum likelihood estimator (NPMLE) based on $n$ independent and identically distributed samples. We show that the unconstrained NPMLE achieves the minimax optimal rate $\\log(n)/n$ in squared Hellinger distance, when the true mixing measure is sub-Gaussian. In addition, we show that when the true mixing measure has compact support, the NMPLE constrained to a compact interval also achieves the minimax optimal rate, which is slightly faster and of order $\\log(n)/(n \\log \\log n)$. Our proof consists of sharply characterizing the local Hellinger bracketing entropy of the class of finite Gaussian mixtures. One of our key technical innovations is to sharply bound the $L^2$ bracketing entropy of balls of bounded-degree polynomials, which may be of independent interest. As an application, we establish the regret-optimality of $g$-modeling procedures based on the NPMLE in the Gaussian Empirical Bayes problem with bounded priors.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30109",
  "author_total": 2,
  "authors": [
    {
      "name": "Shayan Hundrieser"
    },
    {
      "name": "Tudor Manole"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84vhjqdy5vt9b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30109",
  "fulltext_url": null,
  "id": "84vhjqdy5vt9b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30109",
  "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",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
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      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.ST"
    ],
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      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Bayesian Methods and Mixture Models"
    ]
  ],
  "openalex": "W7223252578",
  "pdf_url": "https://arxiv.org/pdf/2610.30109",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84vhjqdy5vt9b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.30109",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30109"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance",
  "title_text": "Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:29Z",
  "url": "https://research.draconitic.com/paper/84vhjqdy5vt9b",
  "venue": "arXiv",
  "versions": []
}