Estimation of Hüsler–Reiss Distributions and Brown–Resnick Processes

Abstract

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(t 0) conditional on η(t 0) exceeding a high threshold, where t 0 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.