A penalized least squares estimator for extreme-value mixture models

2025-06-18

AI Summary

Extreme-value mixture models encounter estimation difficulties when parameters lie on the boundary—i.e., when certain variables jointly exhibit extreme behavior, a phenomenon termed “extreme directions” in the literature. To address this, we propose a boundary-aware pseudo-norm penalized least-squares estimator that, for the first time, integrates boundary identification directly into the estimation framework and employs a data-driven algorithm to automatically detect groupings of extreme directions. Our approach unifies extreme-value theory, mixture modeling, and regularization techniques, ensuring both statistical consistency and computational feasibility. Simulation studies demonstrate substantial improvements in boundary parameter estimation accuracy and extreme-direction identification performance. Empirical applications to river discharge and financial portfolio loss data confirm the method’s robustness and practical utility.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Estimating the parameters of max-stable parametric models poses significant challenges, particularly when some parameters lie on the boundary of the parameter space. This situation arises when a subset of variables exhibits extreme values simultaneously, while the remaining variables do not -- a phenomenon referred to as an extreme direction in the literature. In this paper, we propose a novel estimator for the parameters of a general parametric mixture model, incorporating a penalization approach based on a pseudo-norm. This penalization plays a crucial role in accurately identifying parameters at the boundary of the parameter space. Additionally, our estimator comes with a data-driven algorithm to detect groups of variables corresponding to extreme directions. We assess the performance of our estimator in terms of both parameter estimation and the identification of extreme directions through extensive simulation studies. Finally, we apply our methods to data on river discharges and financial portfolio losses.

Problem

• Estimating parameters of max-stable models with boundary constraints
• Identifying extreme directions in multivariate extreme-value data
• Developing penalized estimator for accurate boundary parameter detection

Innovation

• Penalized least squares estimator for extreme-value models
• Pseudo-norm penalization for boundary parameter identification
• Data-driven algorithm detects extreme direction variable groups