Strong Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in r-Largest Generalized Extreme-Value Models
Abstract
We study maximum likelihood estimation in the three-parameter r-largest generalized extreme-value (rGEV) model with a fixed number r of upper order statistics per block. Because the support depends on all three parameters, the classical regular theory does not apply. We show that the maximum likelihood estimator is strongly consistent on compact parameter sets contained in k<1, where k=−γ denotes the shape parameter. For k0<1/2, the range in which the model is differentiable in quadratic mean, the estimator is root-n consistent, asymptotically normal and asymptotically efficient. The proof verifies the conditions of the moving-support likelihood theorem of Bücher and Segers for the ordered r-dimensional likelihood; the main new ingredients are a sequential modified likelihood criterion and a support-loss bound showing that the regularity threshold k0<1/2 does not depend on r. In simulated heavy-tailed cases with only $50$ blocks, using five to seven order statistics instead of the block maximum reduces the relative bias of the 100-year return level from $6−−13\%$ to below 5%.
Related
- Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator
- Asymptotic Normality of the Hill Estimator in Linear Preferential Attachment
- Existence and consistency of weighted maximum likelihood estimator for extreme quantile regression
- Asymptotic Theory for Restricted Maximum Likelihood Estimators in Generalized Linear Mixed Models
- Combinatorial Insights into Asymptotic Normality
BibTeX
@misc{park2026strong,
title = {{Strong Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in $r$-Largest Generalized Extreme-Value Models}},
author = {Jihong Park and Yire Shin and Jeong-Soo Park},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21161},
eprint = {2610.21161},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21161},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Strong Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in r-Largest Generalized Extreme-Value Models
AU - Jihong Park
AU - Yire Shin
AU - Jeong-Soo Park
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21161
UR - https://arxiv.org/abs/2610.21161
AB - We study maximum likelihood estimation in the three-parameter r-largest generalized extreme-value (rGEV) model with a fixed number r of upper order statistics per block. Because the support depends on all three parameters, the classical regular theory does not apply. We show that the maximum likelihood estimator is strongly consistent on compact parameter sets contained in k<1, where k=−γ denotes the shape parameter. For k_0<1/2, the range in which the model is differentiable in quadratic mean, the estimator is root-n consistent, asymptotically normal and asymptotically efficient. The proof verifies the conditions of the moving-support likelihood theorem of Bücher and Segers for the ordered r-dimensional likelihood; the main new ingredients are a sequential modified likelihood criterion and a support-loss bound showing that the regularity threshold k_0<1/2 does not depend on r. In simulated heavy-tailed cases with only $50$ blocks, using five to seven order statistics instead of the block maximum reduces the relative bias of the 100-year return level from $6−−13\%$ to below 5%.
ER -