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Table 1 Equations of copula functions. Here, u and ν are two dependent univariate variables, df is the degree of freedom, θ and ρ are copula dependence parameters, ∅ is the CDF of standard univariate Gaussian distribution, and tdf is the Student t distribution function

From: Climate change impact assessment on mild and extreme drought events using copulas over Ankara, Turkey

Function (family) Joint CDF, C(u, v) Parameter range
Clayton (Archimedean) [max(0, u−θ + v−θ − 1)]−1/θ 0 ≤ θ
Frank (Archimedean) \( \frac{-1}{\uptheta}\ln \left(1+\frac{\left({\mathrm{e}}^{-\uptheta \mathrm{u}}-1\right)\left({\mathrm{e}}^{-\uptheta \mathrm{v}}-1\right)}{{\mathrm{e}}^{-\uptheta}-1}\right) \) θ ≠ 0
Gumbel Hougaard (Archimedean) \( {\mathrm{e}}^{{\left[{\left(-\ln \mathrm{u}\right)}^{\uptheta}+{\left(-\ln \mathrm{v}\right)}^{\uptheta}\right]}^{\frac{1}{\uptheta}}} \) 1 ≤ θ
Joe (Archimedean) \( 1-{\left({\left(1-\mathrm{u}\right)}^{\uptheta}+{\left(1-\mathrm{v}\right)}^{\uptheta}-{\left(1-\mathrm{u}\right)}^{\uptheta}{\left(1-\mathrm{v}\right)}^{\uptheta}\right)}^{\frac{1}{\uptheta}} \) 1 ≤ θ
Normal (Elliptical) \( \underset{-\infty }{\overset{\varnothing^{-1}\left(\mathrm{u}\right)}{\int }}\underset{-\infty }{\overset{\varnothing^{-1}\left(\mathrm{v}\right)}{\int }}\frac{1}{2\uppi {\left(1-{\uprho}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\mathrm{e}}^{-\frac{{\mathrm{u}}^2-2\uprho \mathrm{uv}+{\mathrm{v}}^2}{2\left(1-{\uprho}^2\right)}}\mathrm{dudv} \) −1 ≤ ρ ≤ 1
Student t (Elliptical) \( \underset{-\infty }{\overset{{{\mathrm{t}}_{\mathrm{df}}}^{-1}\left(\mathrm{u}\right)}{\int }}\underset{-\infty }{\overset{{{\mathrm{t}}_{\mathrm{df}}}^{-1}\left(\mathrm{v}\right)}{\int }}\frac{1}{2\uppi {\left(1-{\uprho}^2\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{\left(1+\frac{{\mathrm{u}}^2-2\uprho \mathrm{uv}+{\mathrm{v}}^2}{\mathrm{df}\left(1-{\uprho}^2\right)}\right)}^{-\frac{\mathrm{df}+2}{2}}\mathrm{dudv} \) −1 ≤ ρ ≤ 1 df ≥ 1