Families of copulas closed under the construction of generalized linear means
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 generation 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.
|Date of creation:||2011|
|Date of revision:|
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- Matthias Fischer & Ingo Klein, 2007. "Constructing Generalized FGM Copulas by Means of Certain Univariate Distributions," Metrika, Springer, vol. 65(2), pages 243-260, February.
- Klein, Ingo & Fischer, Matthias J. & Pleier, Thomas, 2011. "Weighted power mean copulas: Theory and application," IWQW Discussion Paper Series 01/2011, Friedrich-Alexander-Universität Erlangen-Nürnberg, Institut für Wirtschaftspolitik und Quantitative Wirtschaftsforschung (IWQW).
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