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.

翻译：描述$d$维随机向量$X$极值相依性的常用对象是稳定尾部相依函数$L$。目前已出现多种参数模型，其中一个流行的子类由那些产生于具有重尾因子的线性和最大线性因子模型的稳定尾部相依函数构成。此时稳定尾部相依函数由$d \times K$矩阵$A$参数化，其中$K$表示因子数量，$A$可解释为因子载荷矩阵。我们在对$A$附加称为“纯变量假设”的条件下研究$L$的估计问题。将$K \in \{1, \dots, d\}$和$A \in [0, \infty)^{d \times K}$均视为未知参数，这构成了非常规的参数空间，无法纳入常规的估计框架。我们提出两种能够估计$K$和$A$的算法，并为两种算法提供有限样本保证。值得注意的是，这些保证允许维度$d$大于样本量$n$的情况。通过数值实验和两个案例研究对结果进行了说明。