3.2.5. Copulas Approaches

According to Gudendorf and Segers (2010), Copulas refers to specific multivariate distribution functions, in which the distribution function *H* of a n-dimensional random vector *X* = (*<sup>X</sup>*1,..., *Xd*) is the function defined by

$$H(\mathbf{x}) = P(X \le \mathbf{x}) = P(X\_1 \le x\_1, \dots, X\_d \le x\_d) \tag{8}$$

with *x* = (*<sup>x</sup>*1,..., *xd*) ∈ R*d*.

The distribution function *Fj* of *Xj* with *j* ∈ {1, . . . , *d*} can be recalled by the multivariate distribution function *H* by *Fjxj* = *<sup>H</sup>*<sup>∞</sup>,..., <sup>∞</sup>, *xj*, <sup>∞</sup>,..., <sup>∞</sup>, *xj* ∈ R. Therefore, *Fj*, ... , *Fd* is also known as univariate margins of *H* (or called as marginal distribution functions of *X*). One of the concise Copulas definitions is a multivariate distribution function with standard uniform univariate margins, that is, U(0, 1) margins.


Malevergne and Sornette (2006) also asserted that the parameter for dependence structure *Rg* can be estimated as follows

$$\hat{R}\_{\mathcal{J}} = \operatorname\*{argmax}\_{R\_{\mathcal{J}}} \sum\_{i=1}^{n} \log c\_{\mathcal{J}}(\check{\mathbf{x}}\_{i}; R\_{\mathcal{J}}) \tag{9}$$

In which, *x*- is considered as increasing transformations of variable *x*. Then, random vector *x*- and *x* share the same Copulas function. The method of maximum likelihood estimates this parameter in Equation (5). To be more specific, *cgx*-, *Rg* denotes the density function of the Gaussian Copulas with parameter *Rg*.
