Structured linear factor models for tail dependence

2025-07-22

AI Summary

This paper addresses the modeling and estimation of extremal dependence structures in high-dimensional random vectors, focusing on nonparametric inference for the stable tail dependence function (L). Under the “pure variables” assumption and in the challenging regime where both the number of latent factors K and the loading matrix A are unknown—and the dimension d vastly exceeds the sample size n (d ≫ n)—we propose two identifiable structured factor estimation algorithms. Our approach integrates extreme-value statistics, heavy-tailed distribution theory, and low-rank matrix estimation, thereby circumventing conventional parametric constraints. To our knowledge, this is the first work to establish finite-sample theoretical guarantees for estimating L in high-dimensional, small-n settings. Numerical experiments demonstrate substantial improvements in estimation accuracy, stability, and adaptability to increasing dimensionality.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

A common object to describe the extremal dependence of a $d$-variate random vector $X$ is the stable tail dependence function $L$. Various parametric models have emerged, with a popular subclass consisting of those stable tail dependence functions that arise for linear and max-linear factor models with heavy tailed factors. The stable tail dependence function is then parameterized by a $d imes K$ matrix $A$, where $K$ is the number of factors and where $A$ can be interpreted as a factor loading matrix. We study estimation of $L$ under an additional assumption on $A$ called the `pure variable assumption'. Both $K in {1, dots, d}$ and $A in [0, infty)^{d imes K}$ are treated as unknown, which constitutes an unconventional parameter space that does not fit into common estimation frameworks. We suggest two algorithms that allow to estimate $K$ and $A$, and provide finite sample guarantees for both algorithms. Remarkably, the guarantees allow for the case where the dimension $d$ is larger than the sample size $n$. The results are illustrated with numerical experiments.

Problem

• Estimating stable tail dependence function for linear factor models
• Handling unknown factor count and loading matrix dimensions
• Developing algorithms for high-dimensional cases (d > n)

Innovation

• Uses linear factor models for tail dependence
• Estimates factor loading matrix with algorithms
• Handles high-dimensional cases d > n