Modelling the dependence structure of multivariate and spatial extremes

Abstract

Projection of future extreme events is a major issue in a large number of areas including the environment and risk management. Although univariate extreme value theory is well understood, there is an increase in complexity when trying to understand the joint extreme behaviour between two or more variables. Particular interest is given to events that are spatial by nature and which define the context of infinite dimensions. Under the assumption that events correspond marginally to univariate extremes, the main focus is then on the dependence structure that links them.

First, we provide a review of parametric dependence models in the multivariate framework and illustrate different estimation strategies. The spatial extension of multivariate extremes is introduced through max-stable processes. We derive the finite-dimensional distribution of the widely used Brown-Resnick model which permits inference via full and composite likelihood methods.

We then use Skew-symmetric distributions to develop a spectral representation of a wider max-stable model: the extremal Skew-t model from which most models available in the literature can be recovered. This model has the nice advantages of exhibiting skewness and non-stationarity, two properties often held by environmental spatial events. The latter enables a larger spectrum of dependence structures. Indicators of extremal dependence can be calculated using its finite-dimensional distribution.

Finally, we introduce a kernel based non-parametric estimation procedure for univariate and multivariate tail density and apply it for model selection. Our method is illustrated by the example of selection of physical climate models.