Conditional copulas Hi,
As part of one of my courses I need to simulate Gaussian, Student-T and Clayton copulas. The only way to do it that I am aware of uses the conditional copulas, so
$$C_{1|2}(u, v) = P[X = F_1^{-1}(u) | Y = F_2^{-1}(v)]$$
What are the formulae for conditional copulas for Gaussian, Student-T and Clayton?
 A: I understand that you want to generate and plot samples of Gaussian, t and Clayton copulas.
The method of sampling is different for the first two cases (elliptical copula) and the
third case (Archimedean copula).
Here is a brief description of the sampling. A reference for both cases is given:
In the first two cases, it is unnecessary (and not useful) to write a closed formula for
the copula density function. The algorithm for sampling the copula is as follows:
For the Gaussian case:


*

*Generate two random vectors X1 and X2 from a multivariate Gaussian distribution having a given mean, correlation coefficient rho and standard deviation sigma. 

*Compute the marginal cumulative univariate Gaussian distribution functions 
(U1 and U2) corresponding to each vector of samples (X1 and X2) given the appropriate mean and sigma.

*Plot the samples of U2 as a function of U1
For the t-case, the first step is replaced by a multivariate t-distribution with given 
correlation coefficient rho and given number of degrees of freedom v, and the second step 
is replaced by cumulative univariate t-distribution with v degrees of freedom.
Here is a Mathworks reference showing examples of the upper two cases using Matlab functions from the statistics toolbox.
For the third case (Clayton), the algorithm is as follows:


*

*Generate a random vector V from a Gamma distribution with the parameters (1/theta, 1).

*Generate two uniform random independent vectors X1 and X2

*The copula samples are computed as: U1 = psi(-ln(X1)/V) and U2 = psi(-ln(X2)/V), where psi 
is the Clayton generator.


Here is the article by: Alexander J. McNeil, where this algorithm is described.
