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Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design — arXiv:2609.34757

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.34757
License (as archived): CC0-1.0
Archived: 2026-10-07T23:23:21.364Z
SHA-256 of the archived text: 389e980e293ce5f6fd7efef9dce1f59237826d56d253ccdaecfeeff2764ceebd
Stable URL of this page: https://metraipolis.com/documento/dc_c47b032e75cf

How to cite

Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design — arXiv:2609.34757 — arXiv (en), https://arxiv.org/abs/2609.34757 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_c47b032e75cf (Metraipolis archive dc_c47b032e75cf, sha256 389e980e293ce5f6fd7efef9dce1f59237826d56d253ccdaecfeeff2764ceebd)

BibTeX

@misc{dcc47b032e75cf,
  title = {Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design — arXiv:2609.34757},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.34757},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_c47b032e75cf; sha256 389e980e293ce5f6fd7efef9dce1f59237826d56d253ccdaecfeeff2764ceebd},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.34757. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the maximum likelihood estimation of the coefficient {\beta} 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 {\beta} 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+\epsilon} term with {\epsilon} \in (0, 1) by providing an upper bound on the distance between the MLE and {\beta} whenever n > d^{1+\epsilon}. 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-29 in math.ST).
Categorie arXiv: Statistics Theory (math.ST).
Pagina del lavoro: https://arxiv.org/abs/2609.34757 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T23:02:15.117Z dalla pagina https://arxiv.org/catchup/math.ST/2026-09-29?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).