An alternative point process framework for modelling multivariate extreme values

Classical techniques for analysing multivariate extremes can often be framed in terms of the point process representation of de Haan (1985). Amongst other things, this representation provides a characterisation of the limiting distribution of the normalised componentwise maxima of independent and identically distributed unit Fréchet variables, i.e. the class of multivariate extreme value distributions. The dependence structures accommodated within this class correspond only to asymptotic dependence or to exact independence, and so are rather restrictive. In this paper, an alternative limiting point process representation is studied that holds regardless of whether the underlying data generation mechanism is asymptotically dependent or asymptotically independent. Through the use of the usual pseudo-polar coordinates, we characterise the intensity function of this point process in terms of the coefficient of tail dependence η ∈ (0, 1] and a non-negative measure that has to satisfy a simple normalisation condition but is otherwise arbitrary. We use this point process representation to derive an analogue of the standard componentwise maxima result that holds for both asymptotically dependent and asymptotically independent cases. We illustrate our results using a flexible parametric example and provide methods for simulating from both the limiting point process and the limiting componentwise maxima distribution.