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Unter verallgemeinerter Mittelwertbildung abgeschlossene Familien von Copulas

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  • Klein, Ingo

Abstract

We will identify sufficient and partly necessary conditions for a family of copulas to be closed under the construction of generalized linear mean values. These families of copulas generalize results well-known from the literature for the Farlie-Gumbel-Morgenstern (FGM), the Ali-Mikhai-Haq (AMH) and the Barnett-Gumbel (BG) families of copulas closed under weighted linear, harmonic and geometric mean. For these generalizations we calculate the range of Spearman's ρ depending on the choice of weights α, the copulas generating function φ and the exponent γ determining what kind of mean value will be considered. It seems that FGM and AMH generating function φ(υ) = 1 - υ maximizes the range of Spearman's ρ. Furthermore, it will be shown that the considered families of copulas closed under the construction of generalized linear means have no tail dependence in the sense of Ledford & Tawn.

Suggested Citation

  • Klein, Ingo, 2010. "Unter verallgemeinerter Mittelwertbildung abgeschlossene Familien von Copulas," Discussion Papers 86/2010, Friedrich-Alexander University Erlangen-Nuremberg, Chair of Statistics and Econometrics.
    • Handle: RePEc:zbw:faucse:862010
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    References listed on IDEAS

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    1. Matthias Fischer & Ingo Klein, 2007. "Constructing Generalized FGM Copulas by Means of Certain Univariate Distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 65(2), pages 243-260, February.
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    Keywords

    copula; generalized linear means; Spearman's ρ; tail dependence;
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