Structured linear factor models for tail dependence

Abstract A common object to describe the extremal dependence of a d -variate random vector $$\varvec{X}$$ 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 $$d \times K$$ 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 $$K \in \{1, \dots , d\}$$ K ∈ { 1 , ⋯ , d } and $$A \in [0, \infty )^{d \times K}$$ 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 for estimation of 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.