Table 1 Characteristics of Alternative Copula Structures† range and θ Dependence Structure Archimedean Copula value for and range Spearman’s and range ’s Characteristics Generation Function index Radially symmetric, weak 1 –1 ≤ θ ≤ tail dependencies, left and Gaussian Not applicable Not applicable = 0 is θ right tail dependencies go to independence zero at extremes Radially symmetric, only 1 –1 ≤ θ ≤ FGM moderate dependencies can Not applicable Not applicable = 0 is θ be accommodated independence Radially asymmetric, strong left tail dependence and weak 0 < < θ ∞ Clayton tail dependence, right 0 is No simple form θ → tail dependence goes to zero independence at right extreme Radially asymmetric, weak left tail dependence, strong 1 < ≤ θ ∞ Gumbel tail dependence, left tail = 1 is No simple form θ dependence goes to zero at independence left extreme Radially symmetric, very weak tail dependencies (even < < –∞ θ ∞ See Equation Frank weaker than Gaussian), left 0 is * θ → (22) and right tail dependencies independence go to zero at extremes Radially asymmetric, weak left tail dependence and very 1 < ≤ θ ∞ strong right tail dependence See Equation Joe = 1 is No simple form θ (stronger than Gumbel), left (24) independence tail dependence goes to zero at left extreme * Bhat and Eluru (2009) †Source:
Table 1 Characteristics of Alternative Copula Structures† Copula Dependence Structure Characteristics Archimedean Generation Function θ range and value for index ’s and range Spearman’s and range Gaussian Radially symmetric, weak tail dependencies, left and right tail dependencies go to zero at extremes Not applicable Not applicable –1 ≤ θ ≤ 1 θ = 0 is independence FGM Radially symmetric, only moderate dependencies can be accommodated Not applicable Not applicable –1 ≤ θ ≤ 1 θ = 0 is independence Clayton Radially asymmetric, strong left tail dependence and weak tail dependence, right tail dependence goes to zero at right extreme 0 < θ < ∞ θ → 0 is independence No simple form Gumbel Radially asymmetric, weak left tail dependence, strong tail dependence, left tail dependence goes to zero at left extreme 1 ≤ θ < ∞ θ = 1 is independence No simple form Frank Radially symmetric, very weak tail dependencies (even weaker than Gaussian), left and right tail dependencies go to zero at extremes –∞ < θ < ∞ θ → 0 is independence See Equation (22) * Joe Radially asymmetric, weak left tail dependence and very strong right tail dependence (stronger than Gumbel), left tail dependence goes to zero at left extreme 1 ≤ θ < ∞ θ = 1 is independence See Equation (24) No simple form * †Source: Bhat and Eluru (2009)
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