Estimating Max-Stable Random Vectors with Discrete Spectral Measure using Model-Based Clustering

Abstract

This study introduces a novel estimation method for the entries and structure of a matrix $A$ in the linear factor model $\mathbf{X} = A\textbf{Z} + \textbf{E}$. This is applied to an observable vector $\mathbf{X} \in \mathbb{R}^d$ with $\textbf{Z} \in \mathbb{R}^K$, a vector composed of independently regularly varying random variables, and lighter tail noise $\textbf{E} \in \mathbb{R}^d$. The spectral measure of the regularly varying random vector $\mathbf{X}$ is subsequently discrete and completely characterised by the matrix $A$. It follows that the behaviour of its maxima can be modelled by a max-stable random vector with discrete spectral measure. Every max-stable random vector with discrete spectral measure can be written as a linear factor 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 $\mathbf{X}$ 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 $\mathbf{X}$. We apply the suggested method to weekly maxima rainfall and wildfires to illustrate its applicability.