Extremes

, Volume 19, Issue 3, pp 405–427 | Cite as

Mass distributions of two-dimensional extreme-value copulas and related results

  • Wolfgang Trutschnig
  • Manuela Schreyer
  • Juan Fernández-Sánchez
Article

Abstract

Working with Markov kernels (conditional distributions) and right-hand derivatives D + A of Pickands dependence functions A we study the way two-dimensional extreme-value copulas (EVCs) C A distribute mass. Underlining the usefulness of working directly with D + A, we give first an alternative simple proof of the fact that EVCs with piecewise linear A can be expressed as weighted geometric mean of some EVCs whose dependence functions A have at most two edges and present a generalization of this result. After showing that the discrete component of the Markov kernel of C A concentrates its mass on the graphs of some increasing homeomorphisms f t , we determine which EVC assigns maximum mass to the union of the graphs of \(f_{t_{1}},\ldots ,f_{t_{N}}\), derive the absolutely continuous component of an arbitrary EVC C A and deduce that the minimum copula M is the only (purely) singular EVC. Additionally, we prove the existence of EVCs C A which, despite their simple analytic form, exhibit the following surprisingly singular behavior: the discrete, the absolutely continuous and the singular component of the Lebesgue decomposition of the Markov kernel \(K_{C_{A}}(x,\cdot )\) of C A have full support [0,1] for every x∈[0,1].

Keywords

Extreme-value copula Pickands dependence function Markov kernel Singular measure Extreme points 

AMS 2000 Subject Classifications

62H20 62G32 60E05 26A30 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Wolfgang Trutschnig
    • 1
  • Manuela Schreyer
    • 1
  • Juan Fernández-Sánchez
    • 2
  1. 1.Department for MathematicsUniversity SalzburgSalzburgAustria
  2. 2.Grupo de Investigación de Análisis MatemáticoUniversidad de AlmeríaAlmeríaSpain

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