Structured linear factor models for tail dependence

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 × 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 ∈ \1, , d\ and A ∈ [0, ∞)d × 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.