Multivariate peaks over thresholds models

Multivariate peaks over thresholds modeling based on generalized Pareto distributions has up to now only been used in few and mostly low-dimensional situations. This paper contributes to the theoretical understanding, physically based models, inference tools, and simulation methods needed to support routine use, also in high dimensions. We derive a general model for extreme episodes in data, and show how conditioning the distribution of extreme episodes on threshold exceedance gives three basic representations of the family of generalized Pareto distributions. The first representation is constructed on the real scale of the observations. The second one starts with a model on a standard exponential scale which then is transformed to the real scale. The third is a spectral representation proposed in A. Ferreira and L. de Haan [Bernoulli 20 (2014) 1717-1737]. Numerically tractable forms of densities and censored densities are derived, and give tools for exible parametric likelihood inference. Our results are aimed at statistical analysis of complex and multidimensional extreme episodes, such as simultaneous ooding of many dykes; landslides caused by heavy rainfall over one or several days; or deaths caused by a sequence of very hot nights during a heat wave.