Unrestricted conditional maximum likelihood estimation for a Fréchet regression model
Abstract
Heavy-tailed response variables are routinely modeled as a function of covariates, but the asymptotic theory available for the resulting maximum likelihood estimators sidesteps certain technical difficulties by restricting the parameters to a compact set, which is not the natural parameter space. We consider a Fréchet model whose scale parameter depends log-linearly on a vector of covariates, while the tail parameter is constant, and we study the conditional maximum likelihood estimator of the tail parameter and the regression coefficients jointly. We show that this estimator is consistent over the full, noncompact parameter space, and that it is asymptotically normal, with the inverse Fisher information as limiting covariance matrix. Consistency is obtained by a specific compactification of the parameter space relying on the cosmic closure of Euclidean space. The extension of Wald's consistency theorem needed for that step, from single observations to blocks of observations, is stated in an abstract setting and is possibly of independent interest.
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BibTeX
@misc{reverbel2026unrestricted,
title = {{Unrestricted conditional maximum likelihood estimation for a Fréchet regression model}},
author = {Marcio Reverbel and Johan Segers},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14607},
eprint = {2609.14607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14607},
}
RIS
TY - UNPB
TI - Unrestricted conditional maximum likelihood estimation for a Fréchet regression model
AU - Marcio Reverbel
AU - Johan Segers
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14607
UR - https://arxiv.org/abs/2609.14607
AB - Heavy-tailed response variables are routinely modeled as a function of covariates, but the asymptotic theory available for the resulting maximum likelihood estimators sidesteps certain technical difficulties by restricting the parameters to a compact set, which is not the natural parameter space. We consider a Fréchet model whose scale parameter depends log-linearly on a vector of covariates, while the tail parameter is constant, and we study the conditional maximum likelihood estimator of the tail parameter and the regression coefficients jointly. We show that this estimator is consistent over the full, noncompact parameter space, and that it is asymptotically normal, with the inverse Fisher information as limiting covariance matrix. Consistency is obtained by a specific compactification of the parameter space relying on the cosmic closure of Euclidean space. The extension of Wald's consistency theorem needed for that step, from single observations to blocks of observations, is stated in an abstract setting and is possibly of independent interest.
ER -