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Expansions of multivariate Pickands densities and testing the tail dependence

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  • Frick, Melanie
  • Reiss, Rolf-Dieter

Abstract

Multivariate extreme value distribution functions (EVDs) with standard reverse exponential margins and the pertaining multivariate generalized Pareto distribution functions (GPDs) can be parametrized in terms of their Pickands dependence function D with D=1 representing tail independence. Otherwise, one has to deal with tail dependence. Besides GPDs we include in our statistical model certain distribution functions (dfs) which deviate from the GPDs whereby EVDs serve as special cases. Our aim is to test tail dependence against rates of tail independence based on the radial component. For that purpose we study expansions and introduce a second order condition for the density (called Pickands density) of the joint distribution of the angular and radial component with the Pickands densities under GPDs as leading terms. A uniformly most powerful test procedure is established based on asymptotic distributions of radial components. It is argued that there is no loss of information if the angular component is omitted in the testing problem.

Suggested Citation

  • Frick, Melanie & Reiss, Rolf-Dieter, 2009. "Expansions of multivariate Pickands densities and testing the tail dependence," Journal of Multivariate Analysis, Elsevier, vol. 100(6), pages 1168-1181, July.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:6:p:1168-1181
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    References listed on IDEAS

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    1. Falk, Michael & Reiss, Rolf-Dieter, 2005. "On the distribution of Pickands coordinates in bivariate EV and GP models," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 267-295, April.
    2. Falk, Michael & Reiss, Rolf-Dieter, 2005. "On Pickands coordinates in arbitrary dimensions," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 426-453, February.
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    Cited by:

    1. Keef, Caroline & Papastathopoulos, Ioannis & Tawn, Jonathan A., 2013. "Estimation of the conditional distribution of a multivariate variable given that one of its components is large: Additional constraints for the Heffernan and Tawn model," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 396-404.
    2. Melanie Frick, 2012. "Measures of multivariate asymptotic dependence and their relation to spectral expansions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(6), pages 819-831, August.
    3. Frick, Melanie & Reiss, Rolf-Dieter, 2010. "Limiting distributions of maxima under triangular schemes," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2346-2357, November.
    4. Liu, Y. & Tawn, J.A., 2014. "Self-consistent estimation of conditional multivariate extreme value distributions," Journal of Multivariate Analysis, Elsevier, vol. 127(C), pages 19-35.

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