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

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{é}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.

arXiv comment37 pages, 18 Figures and 3 Tables

References in corpus (6)

• Space-time modelling of extreme events
• Extremal t processes: Elliptical domain of attraction and a spectral representation
• Extremes on river networks
• On the asymptotic joint distribution of sample space--time covariance estimators
• Bootstrapping Empirical Processes of Cluster Functionals with Application to Extremograms
• Limit theory for the empirical extremogram of random fields