*5.4. Gumbel Copulas*

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

$$\mathcal{C}\_{\theta}(\boldsymbol{\mu}, \boldsymbol{\upsilon}) = \exp \left( - \left[ (-\ln \boldsymbol{\mu})^{\theta} + (-\ln \boldsymbol{\upsilon})^{\theta} \right] \overline{\overset{\theta}{\theta}}{\boldsymbol{\theta}} \right) , \quad \theta > 0.$$

The parameter *θ* = 1 implies uncorrelated (*<sup>X</sup>*1, *<sup>X</sup>*2). Figures 7 and 8 show the behavior of *Y* and *Z* via the copulas. Tables 7 and 8 represent some estimated percentiles for both *Y* and *Z*. In the comparison with Clayton, Gumbel Copula gets left skewness, higher median, and less spread for *Y*, but gets a symmetric shape, unchanged median, less spread, and smaller scale (higher spike) for *Z*. However, the shape of *Y* tends to be more symmetric when one increases the parameter *θ*.

**Table 7.** Some percentiles of *Y* = *X*1/*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Gumbel Copulas.


**Table 8.** 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 Gumbel Copulas.


**Figure 7.** PDFs and CDFs of the ratio *Y* = *X*1 *X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Gumbel Copulas.

**Figure 8.** PDFs and CDFs of the ratio *Z* = *X*1 *X*1+*X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Gumbel Copulas.
