Multivariate peaks over thresholds models

Abstract

Multivariate peaks over thresholds modelling based on generalized Pareto distributions has up to now only been used in few and mostly two-dimensional situations. This paper contributes theoretical understanding, models which can respect physical constraints, inference tools, and simulation methods to support routine use, with an aim at higher dimensions. We derive a general point process model for extreme episodes in data, and show how conditioning the distribution of extreme episodes on threshold exceedance gives four 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 is then transformed to the real scale. The third and fourth representations are reformulations of a spectral representation proposed in Ferreira and de Haan (Bernoulli 20(4), 2014). Numerically tractable forms of densities and censored densities are found and give tools for flexible parametric likelihood inference. New simulation algorithms, explicit formulas for probabilities and conditional probabilities, and conditions which make the conditional distribution of weighted component sums generalized Pareto are derived.

A Spliced Gamma-Generalized Pareto Model for Short-Term Extreme Wind Speed Probabilistic Forecasting

“…We fitted the bulk and the tail of the wind speed distribution separately, as extreme events usually behave differently from low and moderately large events, and therefore only extreme observations may give information about the tail of the distribution (Rootzén et al, 2018).…”

Extremal dependence of random scale constructions

“…Pareto copulas can be identified with so-called extreme value copulas, which arise as the limiting copulas of suitably normalized componentwise maxima; see e.g. Rootzén et al (2018). The next result provides the general form of the tail dependence coefficient for these models.…”

Multivariate generalized Pareto distributions: Parametrizations, representations, and properties

“…For statistical inference, it is highly useful to know the probability density functions of the GP distributions constructed via the methods in Section 4. Most of the results of this section can also be found in Rootzén et al (2017), but for completeness, we give their proofs in Appendix B. Theorem 5.5 is new.…”

“…Our aim is to facilitate manipulation of multivariate GP distributions for the analysis of multivariate extremes. Rootzén et al (2017) revisited multivariate GP distributions with an eye towards modelling. To facilitate the incorporation of physical constraints in the construction Date: May 24, 2017. of GP models, these distributions were connected to a number of point process representations.…”