TY - JOUR

T1 - Multivariate peaks over thresholds models

AU - Rootzen, Holger

AU - Segers, Johan

AU - Wadsworth, Jennifer Lynne

N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s10687-017-0294-4

PY - 2018/3

Y1 - 2018/3

N2 - 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 re-formulations of 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 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.

AB - 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 re-formulations of 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 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.

KW - Extreme values

KW - Multivariate generalized Pareto distribution

KW - Peaks over threshold likelihoods

KW - Simulation of extremes

U2 - 10.1007/s10687-017-0294-4

DO - 10.1007/s10687-017-0294-4

M3 - Journal article

VL - 21

SP - 115

EP - 145

JO - Extremes

JF - Extremes

SN - 1386-1999

IS - 1

ER -