Estimation of Hüsler–Reiss distributions and Brown–Resnick processes

Summary

Estimation of extreme value parameters from observations in the max‐domain of attraction of a multivariate max‐stable distribution commonly uses aggregated data such as block maxima. Multivariate peaks‐over‐threshold methods, in contrast, exploit additional information from the non‐aggregated ‘large’ observations. We introduce an approach based on peaks over thresholds that provides several new estimators for processes η in the max‐domain of attraction of the frequently used Hüsler–Reiss model and its spatial extension: Brown–Resnick processes. The method relies on increments η(·)−η(t₀) conditional on η(t₀) exceeding a high threshold, where t₀ is a fixed location. When the marginals are standardized to the Gumbel distribution, these increments asymptotically form a Gaussian process resulting in computationally simple estimates of the Hüsler–Reiss parameter matrix and particularly enables parametric inference for Brown–Resnick processes based on (high dimensional) multivariate densities. This is a major advantage over composite likelihood methods that are commonly used in spatial extreme value statistics since they rely only on bivariate densities. A simulation study compares the performance of the new estimators with other commonly used methods. As an application, we fit a non‐isotropic Brown–Resnick process to the extremes of 12‐year data of daily wind speed measurements.