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。该模型适用于可观测向量$\textbf{X} \in \mathbb{R}^d$，其中$\textbf{Z} \in \mathbb{R}^K$为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所处理的极大线性模型。极限分布的谱随后呈现离散性，并完全由矩阵$A$表征。任何具有离散谱测度的极值稳定随机向量均可表示为极大线性模型。矩阵$A$的每一行均被设定为兼具尺度化与稀疏性。此外，$K$的数值并非先验已知。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（即通过$A$仅与单一潜在因子相关联的$\textbf{X}$分量）的情况下，矩阵$A$可从极值相关矩阵中重构。我们的可识别性证明具有构造性，为基于$n$个弱相关观测数据确定因子数$K$与矩阵$A$的创新估计方法奠定了基础。我们将所提方法应用于周最大降雨量与野火数据，以验证其适用性。