Convex geometry of max-stable distributions

By Ilya S. Molchanov. 2006

54 citations recorded by the source corpus.

It is shown that max-stable random vectors in [0, ∞ )d with unit Fréchet marginals are in one to one correspondence with convex sets K in [0, ∞ )d called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function Pξ ≤ x of a max-stable random vector ξ with unit Fréchet marginals is determined by the norm of the inverse to x, where all possible norms are given by the support functions of (normalised) max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/0027959433a77282b50f5967983d69410c6293d5

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Citations in the corpus

MAX-STABLE MODELS FOR MULTIVARIATE EXTREMES (2012, background): “The fact that ℓθ is a norm is no coincidence: in a remarkable paper by Molchanov (2008), a characterization is given of all the norms that can give rise to stable tail dependence functions.”