Structured linear factor models for tail dependence

TL;DR

This paper introduces new algorithms for estimating the tail dependence structure in multivariate heavy-tailed data using linear factor models, with guarantees even in high-dimensional settings.

Contribution

It proposes two novel algorithms for estimating the number of factors and factor loadings in linear tail dependence models, with finite sample guarantees applicable when dimension exceeds sample size.

Findings

Algorithms effectively estimate tail dependence in high dimensions.

Finite sample guarantees are provided for the proposed methods.

Numerical experiments and case studies validate the approach.

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 $d \times K$ 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 $A \in [0, \infty)^{d \times K}$ 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 and two case studies.