Copula theory

Conditional density through integration

We can easily calculate:

Hence, for each value $u_{1}$, we can calculate the likelihood of observing $(u_{2},u_{3})$. This way, we already know how several values of $u_{1}$ compare to each other in terms of likelihood: some may be twice as likely as others. Still, we need to translate these likelihoods into a probability measure, so that we need to normalize the values:

Given $A$ we get:

The density of the emerging copula can be derived by integration:

Note:

• $F_{2\vert 1}$: distribution of $X_{2\vert 1}$
• $\overline{F}_{2\vert 1}$: distribution of $U_{2\vert 1}$

Conditional distributions can be hard to derive if they do not align with the density decomposition that was used for the vine:

Univariate Conditional distributions aligning with the chosen density decomposition are easy to infer:

Conditional distributions follow a recursive structure:

Thereby, we define:

Emerging copula example: two-dimensional conditioning set

because of