# Generalised least squares estimation of regularly varying space-time processes based on flexible observation schemes

## TL;DR

This paper introduces a two-step estimation method for extremograms of regularly varying space-time processes, combining asymptotic theory with practical subsampling techniques, and demonstrates its effectiveness on Brown-Resnick models.

## Contribution

It develops a novel two-step estimation approach for extremograms in space-time processes, including asymptotic properties and confidence interval procedures.

## Key findings

- Method performs well for moderate sample sizes
- Provides asymptotic normality results for estimators
- Effective application to Brown-Resnick processes

## Abstract

Regularly varying stochastic processes model extreme dependence between process values at different locations and/or time points. For such processes we propose a two-step parameter estimation of the extremogram, when some part of the domain of interest is fixed and another increasing. We provide conditions for consistency and asymptotic normality of the empirical extremogram centred by a pre-asymptotic version for such observation schemes. For max-stable processes with Fr{\'e}chet margins we provide conditions, such that the empirical extremogram (or a bias-corrected version) centred by its true version is asymptotically normal. In a second step, for a parametric extremogram model, we fit the parameters by generalised least squares estimation and prove consistency and asymptotic normality of the estimates. We propose subsampling procedures to obtain asymptotically correct confidence intervals. Finally, we apply our results to a variety of Brown-Resnick processes. A simulation study shows that the procedure works well also for moderate sample sizes.