Symmetry and Complement Functions of a Copula

Sungur, Engin A.; Çelebioğlu, Salih; Kim, Jong Min

doi:10.1501/commua1_0000000193

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Anti-symmetric Copulas With Odd Spearman Rho and Kendall Tau1

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2015

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“…The Hoeffding-Fréchet upper bound is M(u, v) = min(u, v) and the Hoeffding-Fréchet lower bound is W (u, v) = max(u +v −1, 0). Up to a sign change, the following notion is also used in Sungur et al (2007), Definition 2.1. C(u, v) be any copula.…”

Section: Anti-symmetric Copulas With Odd Spearman Rho and Kendall Taumentioning

confidence: 99%

A comprehensive extension of the FGM copula

Hürlimann¹

2015

Stat Papers

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We consider one-parametric families of copulas for which the complement function for independence satisfies an anti-symmetric property. The Spearman rank correlation and Kendall's tau of an anti-symmetric family of copulas are necessarily odd functions of the parameter. Extending the parameter range of the FGM copula to the whole real line and truncated it from above and below using the Hoeffding-Fréchet bounds generates a comprehensive anti-symmetric extension of the FGM copula. The detailed analytical representation of the extended FGM copula, the absolutely continuous and singular components, as well as the Spearman rank correlation and Kendall's tau dependence functions are derived. Several additional examples illustrate the antisymmetric copula construction.

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Section: Anti-symmetric Copulas With Odd Spearman Rho and Kendall Taumentioning

confidence: 99%