*5.5. Frank Copulas*

In addition, we investigate dependence structures of *X*1 and *X*2 through the following Frank Copulas *Cθ* (*<sup>u</sup>*, *v*) and observe the shapes of the corresponding distributions of both *Y* and *Z*:

$$\mathbb{C}\_{\theta}(u,v) = -\frac{1}{\theta} \ln \left( 1 + \frac{\left(\varepsilon^{-\theta u} - 1\right) \left(\varepsilon^{-\theta v} - 1\right)}{\varepsilon^{-\theta} - 1} \right), \quad \theta \in \mathbb{R} \backslash \{0\}.$$

For Frank Copulas, the parameter *θ* represents two independent random variables when it tends to zero, becomes more monotonic structure when *θ* → <sup>∞</sup>, and becomes more counter-monotonic when *θ* → <sup>−</sup>∞. For *θ* = 1, 2, 3, 4, we have Tables 9 and 10, and Figures 9 and 10. In contrast to both Clayton and Gumbel Copulas, the density of *Y* via Frank Copulas is more symmetric and the density of *Z* is not scaling too much, but for the median and spread of *Y*, it behaves like the Gumbel Copulas and Clayton Copulas; that is, if we increase the parameter *θ*, the median also increases, whereas the spread (via IQR) decreases.

**Table 9.** Some percentiles of *Y* = *X*1/*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Frank Copulas.



**Table 10.** Some percentiles of *Y* = *<sup>X</sup>*1/(*<sup>X</sup>*1 + *<sup>X</sup>*2), where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Frank Copulas.

**Figure 9.** PDF and CDF of quotient of the ratio *Y* = *X*1 *X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Frank Copulas.

**Figure 10.** PDF and CDF of the ratio *Z* = *X*1 *X*1+*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Frank Copulas.

*5.6. Joe Copulas*

Finally, we investigate dependence structures of *X*1 and *X*2 through the following Joe Copulas *Cθ* (*<sup>u</sup>*, *v*) and observe the shapes of the corresponding distributions of both *Y* and *Z*:

$$\mathbb{C}\_{\theta}(u,v) = 1 - [(1-u)^{\theta} + (1-v)^{\theta} - (1-u)^{\theta}(1-v)^{\theta}]^{1/\theta}, \quad \theta \in [1,\infty).$$

Similar to Gumbel Copulas, Joe Copula shows independence when *θ* = 1 and becomes more monotonicity if *θ* → ∞. With assistance of the tables and graphs with *θ* = 1, 2, 3, 4, Tables 11 and 12 and Figures 11 and 12 tell us that the distribution behaviors of *Y* are also affected with higher median, less IQR, smaller scale (higher spike), and the shape is more asymmetric if one increases the parameter *θ*. On the other hand, *Z* still gets median unchanged with *median* = 0.5 and less spread with the sum of the first quartile and third quartile is always equal to 1, i.e., (*Q*0.25 + *Q*0.75 = 1).


**Table 11.** Some percentiles of *Y* = *X*1/*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Joe Copulas.

**Table 12.** Some percentiles of *Y* = *<sup>X</sup>*1/(*<sup>X</sup>*1 + *<sup>X</sup>*2), where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Joe Copulas.


**Figure 11.** PDFs and CDFs of the ratio *Y* = *X*1 *X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Joe Copulas.

**Figure 12.** PDFs and CDFs of the ratio *Z* = *X*1 *X*1+*X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Joe Copulas.

*5.7. Comparison of Copulas with the Same Measure of Dependence*

In this section, we investigate the effects of the six copulas families as discussed above on the shapes of different distributions for the random variables *Y* := *X*1/*X*2 and *Z* := *<sup>X</sup>*1/(*<sup>X</sup>*1 + *<sup>X</sup>*2) when they have the same measure of dependence—the Kendall's coefficient *τ*. Here, the parameters are chosen to each copula to correspond to Kendall *τ* = 0.49. We exhibit the corresponding CDFs and PDFs of both *Y* and *Z* in Figures 13 and 14, estimate the percentiles *Qα* for some *α* = 0.05, 0.25, 0.5, 0.75, 0.95, and display the values in Tables 13 and 14. As can be seen from the tables and the figures, *Y* attains the

greatest median (0.76) and the smallest IQR (1.32) with left skewed shape when both *X*1 and *X*2 follow Joe Copula and Student-*t* Copula. In contrast, Gaussian Copula produces the smallest median (0.70) and the largest IQR (1.42) with symmetric shape. Using Clayton Copula, *Y* gets the second biggest median (0.74). The ratio random variable *Y* has the same median (0.72) for both Gumbel and Frank Copula, but the Frank makes *Y* ge<sup>t</sup> higher IQR than the Gumbel (1.41 > 1.33). On the other hand, the random variable *Z* has symmetric shape with unchanged median (0.5) among all investigated copulas. It only changes the scale, where the Joe Copula affects *Z* with the smallest scale (i.e., the tallest height of density) whilst the Frank Copula causes *Z* to ge<sup>t</sup> the greatest scale (i.e., the shortest height of density).

**Table 13.** Some percentiles of *Y* = *X*1/*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) modelled with six copulas having the same Kendall coefficient *τ* = 0.49 .


**Table 14.** Some percentiles of *Y* = *<sup>X</sup>*1/(*<sup>X</sup>*1 + *<sup>X</sup>*2), where (*<sup>X</sup>*1, *<sup>X</sup>*2) modelled with six copulas having the same Kendall coefficient *τ* = 0.49 .


**Figure 13.** PDFs and CDFs of the ratio *Y* = *X*1 *X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) modeled with six Copulas having the same *τ* = 0.49.

**Figure 14.** PDF and CDF of the ratio *Z* = *X*1 *X*1+*X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) modeled with six Copulas having the same *τ* = 0.49.
