{
  "abstract": "We study the maximum likelihood estimation of the coefficient β in well-specified Poisson regression. Using tools from empirical process theory and random conic geometry, we show that the probability of existence of the maximum likelihood estimator (MLE) exhibits a sharp phase transition at the threshold n > d. We then determine a minimum threshold exponential in the norm of β on the sample size n to guarantee with high probability an excess risk of the asymptotic order d/n. We reveal the existence of an intermediate regime in Poisson regression, when n is larger than d but smaller than this exponential threshold, where the MLE exists but does not achieve the optimal rate d/n. We close the gap between the two regimes up to a d^{1+ε} term with ε \\in (0, 1) by providing an upper bound on the distance between the MLE and β whenever n > d^{1+ε}. Along the way, we provide two generalizations of well-known PAC-Bayes inequalities regarding sub-Gamma random vectors and sub-Gamma random matrices that are of independent interest and that we use extensively to prove the main results of the present paper.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34757",
  "author_total": 2,
  "authors": [
    {
      "name": "Matthieu Lerasle"
    },
    {
      "name": "Antoine Roche"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5jg7xt6551b9x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34757",
  "fulltext_url": null,
  "id": "5jg7xt6551b9x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34757",
  "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/2609.34757",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5jg7xt6551b9x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.34757",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34757"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design",
  "title_text": "Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:26Z",
  "url": "https://research.draconitic.com/paper/5jg7xt6551b9x",
  "venue": "arXiv",
  "versions": []
}