Estimating MaxStable Random Vectors with Discrete Spectral Measure using ModelBased Clustering
Abstract
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 maxlinear 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 maxstable random vector with discrete spectral measure can be written as a maxlinear 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2024
 DOI:
 10.48550/arXiv.2402.01609
 arXiv:
 arXiv:2402.01609
 Bibcode:
 2024arXiv240201609B
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 42 pages, 6 figures