This study introduces a novel estimation method for the entries and structure of a matrix $A$ in the linear factor model $\textbf{X} = A\textbf{Z} + \textbf{E}$. This is applied to an observable vector $\textbf{X} \in \mathbb{R}^d$ with $\textbf{Z} \in \mathbb{R}^K$, a vector composed of independently regularly varying random variables, and independent lighter tail noise $\textbf{E} \in \mathbb{R}^d$. This leads to max-linear models treated in classical multivariate extreme value theory. The spectral of the limit distribution is subsequently discrete and completely characterised by the matrix $A$. Every max-stable random vector with discrete spectral measure can be written as a max-linear model. Each row of the matrix $A$ is supposed to be both scaled and sparse. Additionally, the value of $K$ is not known a priori. The problem of identifying the matrix $A$ from its matrix of pairwise extremal correlation is addressed. In the presence of pure variables, which are elements of $\textbf{X}$ linked, through $A$, to a single latent factor, the matrix $A$ can be reconstructed from the extremal correlation matrix. Our proofs of identifiability are constructive and pave the way for our innovative estimation for determining the number of factors $K$ and the matrix $A$ from $n$ weakly dependent observations on $\textbf{X}$. We apply the suggested method to weekly maxima rainfall and wildfires to illustrate its applicability.

翻译：本研究针对线性因子模型 $\textbf{X} = A\textbf{Z} + \textbf{E}$ 中矩阵 $A$ 的项与结构，提出了一种新颖的估计方法。该模型应用于可观测向量 $\textbf{X} \in \mathbb{R}^d$，其中 $\textbf{Z} \in \mathbb{R}^K$ 是由独立正则变化随机变量构成的向量，$\textbf{E} \in \mathbb{R}^d$ 为独立的轻尾噪声向量。这导出了经典多元极值理论中所处理的极大线性模型。其极限分布的谱随后是离散的，并由矩阵 $A$ 完全刻画。任何具有离散谱测度的极大稳定随机向量均可表示为极大线性模型。矩阵 $A$ 的每一行均被假定为经过缩放且稀疏的。此外，$K$ 的值并非先验已知。本文解决了从其成对极值相关矩阵中识别矩阵 $A$ 的问题。在存在纯变量（即 $\textbf{X}$ 中通过 $A$ 仅与单一潜在因子相关联的元素）的情况下，矩阵 $A$ 可从极值相关矩阵中重建。我们的可识别性证明是构造性的，并为基于 $n$ 个弱相关观测数据确定因子数量 $K$ 和矩阵 $A$ 的创新估计方法铺平了道路。我们将所提出的方法应用于周最大降雨量与野火数据，以说明其适用性。