On generalized max-linear models and their statistical interpolation

TL;DR

This paper introduces a generalized max-linear model to generate and interpolate max-stable processes in continuous space from finite observations, enabling effective spatial prediction of extreme events.

Contribution

It extends the max-linear model to generate max-stable processes in continuous space and provides a method for their statistical interpolation from finite data.

Findings

Processes converge uniformly to the original process.

Pointwise mean squared error has a closed-form expression.

Method applies to generalized Pareto processes.

Abstract

We propose a way how to generate a max-stable process in $C[0,1]$ from a max-stable random vector in $\mathbb R^d$ by generalizing the \emph{max-linear model} established by \citet{wansto11}. It turns out that if the random vector follows some finite dimensional distribution of some initial max-stable process, the approximating processes converge uniformly to the original process and the pointwise mean squared error can be represented in a closed form. The obtained results carry over to the case of generalized Pareto processes. The introduced method enables the reconstruction of the initial process only from a finite set of observation points and, thus, reasonable prediction of max-stable processes in space becomes possible. A possible extension to arbitrary dimension is outlined.