*5.2. Student-t Copulas*

We then investigate dependence structures of *X*1 and *X*2 through Student-*t* Copulas *Cr*,*<sup>ν</sup>*(*<sup>u</sup>*, *v*) and observe the shapes of the corresponding distributions of both *Y* and *Z*:

$$\mathbb{C}\_{r,\boldsymbol{v}}(\boldsymbol{u},\boldsymbol{v}) = \frac{1}{2\pi\sqrt{1-\boldsymbol{r}^2}} \int\_{-\infty}^{t\_{\boldsymbol{v}}^{-1}(\boldsymbol{u})} \int\_{-\infty}^{t\_{\boldsymbol{v}}^{-1}(\boldsymbol{v})} \left(1 + \frac{\mathbf{s}^2 - 2\mathbf{r}\mathbf{s}t + t^2}{\boldsymbol{v}(1-\boldsymbol{r}^2)}\right)^{(\boldsymbol{v}+2)/2} d\boldsymbol{s}d\boldsymbol{t} \,\boldsymbol{\omega}$$

where *t*−<sup>1</sup> *ν* (*x*) is the inverse of Student CDF with degrees of freedom *ν*, *r* denotes Pearson's correlation coefficient between *X*1 and *X*2, and |*r*| < 1, and *ν* > 2 is the degrees of freedom. We also consider *r* = −0.9, −0.5, 0, 0.5, 0.9 with three degrees of freedom (*ν* = 3), where *r* = 0 is corresponding to no linear correlation. The PDFs and CDFs of *Y* of *Z* are, respectively, represented in Figures 3 and 4. Some percentiles are estimated and displayed in Tables 3 and 4. Similarly to Gaussian copula, in this case, the center and spread of *Y* and *Z* are also varying in the same way. However, one can see a representation of skewness and tailedness, that is, right skewed if *r* < 0 and left skewed if *r* > 0, since the fact that Student-*t* Copulas can capture tail dependence between *X*1 and *X*2.


**Table 3.** Some percentiles of *Y* = *X*1/*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Student-*t* Copulas, *ν* = 3.

**Table 4.** 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 Student-*t* Copulas, *ν* = 3.


**Figure 3.** PDFs and CDFs of the ratio *Y* = *X*1 *X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Student-*t* Copulas, *ν* = 3.

**Figure 4.** PDFs and CDFs of the ratio *Z* = *X*1 *X*1+*X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Student-*t* Copulas, *ν* = 3. *5.3. Clayton Copulas*

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

$$\mathcal{C}\_{\theta}(\mu, \upsilon) = \max \left\{ \mu^{-\theta} + \upsilon^{-\theta} - 1 \right\}^{-\frac{1}{\theta}}, \ \theta \in [-1; +\infty) \backslash 0.$$

In practice, we use *θ* > 0 that leads to

$$\mathcal{C}\_{\theta}(\mu, \upsilon) = \left(\mu^{-\theta} + \upsilon^{-\theta} - 1\right)^{-} \overset{\text{I}}{\theta}\_{\prime}, \quad \theta > 0.$$

For *θ* = 1, 2, 3, 4, we obtain CDFs and PDFs of *Y* and *Z* and exhibit them in Figures 5 and 6, respectively, and their percentiles as shown in Tables 5 and 6. Clearly, Clayton Copulas affect *Y* to ge<sup>t</sup> heavier left tail and the more positive dependence is; that is, *θ* → <sup>∞</sup>, the greater the median and the smaller the IQR of *Y* tend to be. On the other hand, the shape of distribution of *Z* is still symmetric with unchanged median, less spread, and more spike.

**Table 5.** Some percentiles of *Y* = *X*1/*X*2, where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Clayton Copulas.


**Table 6.** 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 Clayton Copulas.


**Figure 5.** PDFs and CDFs of the ratio *Y* = *X*1 *X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Clayton Copulas.

**Figure 6.** PDFs and CDFs of the ratio *Z* = *X*1 *X*1+*X*2 , where (*<sup>X</sup>*1, *<sup>X</sup>*2) follows Clayton Copulas.
