Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design
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.
Related
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BibTeX
@misc{lerasle2026finite,
title = {{Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design}},
author = {Matthieu Lerasle and Antoine Roche},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34757},
eprint = {2609.34757},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34757},
}
RIS
TY - UNPB
TI - Finite sample behavior of the maximum likelihood estimator in the Poisson model under Gaussian design
AU - Matthieu Lerasle
AU - Antoine Roche
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34757
UR - https://arxiv.org/abs/2609.34757
AB - 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.
ER -