Table 3.

Copula families used in this study and the mathematical descriptions.

Name	Mathematical description	Parameter range	Reference
Gaussian	${\int }_{-\infty }^{{\varnothing }^{-1}\left(u\right)}{\int }_{-\infty }^{{\varnothing }^{-1}\left(v\right)}\frac{1}{2\pi \sqrt{1-{\theta }^2}}exp\left(\frac{2\mathit{\theta xy}-x^2-y^2}{2\left(1-{\theta }^2\right)}\right)\mathrm{dxd}{\mathrm{y}}^b$ θ : Linear correlation coefficient ∅ : Standard normal cumulative distribution function	θ ∈ [−1, 1]	(Renard and Lang, 2007)
t	${\int }_{-\infty }^{t_{{\theta }_2}^{-1}\left(u\right)}{\int }_{-\infty }^{t_{{\theta }_2}^{-1}\left(v\right)}\frac{\mathrm{\Gamma }\left(\frac{{\theta }_2+2}{2}\right)}{\mathrm{\Gamma }\left(\frac{{\theta }_2}{2}\right)\pi {\theta }_2\sqrt{1-{\theta }_1^2}}{\left(1+\frac{x^2-2{\theta }_1\mathit{xy}+y^2}{{\theta }^2}\right)}^{\frac{{\theta }_2+2}{2}}\mathrm{dxd}{\mathrm{y}}^c$ θ1 : Linear correlation coefficient tθ2 : Cumulative distribution function of t distribution with θ2 degree of freedom	θ1 ∈ [−1, 1]; θ2 ∈ [0, ∞]	(Demarta and McNeil, 2005)
Clayton	max(u−θ + v−θ − 1, 0)−1/θ θ : Measure of dependency between u and v.	θ ∈ [−1, ∞]\0	(Clayton, 1978)
Frank	$-\frac{1}{\theta }ln\left[1+\frac{\left(exp\left(-\mathit{\theta u}\right)-1\right)\left(exp\left(-\mathit{\theta v}\right)-1\right)}{exp\left(-\theta \right)-1}\right]$ θ : Similar to Clayton copula	θ ∈ ℝ\0	(Li et al., 2013)
Gumbel	$exp\left\{-{\left[{\left(-\mathit{ln}\left(u\right)\right)}^{\theta }+{\left(-\mathit{ln}\left(v\right)\right)}^{\theta }\right]}^{\frac{1}{\theta }}\right\}$ θ : Similar to Clayton copula	θ ∈ [−1, ∞]	(Zhang and Singh, 2006)