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Tedypendah
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Posts: 55
Joined: May 26th, 2013, 10:11 am

Simulating from Multivariate distributions or Copulas

April 3rd, 2015, 4:55 am

How do we generate random variables from a multivariate distribution other than the multivariate normal distribution? For a multivariate normal distribution you can use Cholesky decomposition with Monte Carlo inverse transform of the standard normal distribution or principal component analysis. I can't see how this could be extended to the case where you have a Multivariate Lognormal distribution for example or Multivariate Poisson.And what if you're using a copula say Gaussian or Archimedean copula like Gumbel? How would one simulate from these?
 
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Samsaveel
Posts: 33
Joined: April 20th, 2008, 5:47 am

Simulating from Multivariate distributions or Copulas

April 3rd, 2015, 1:45 pm

the choice of copula is independent from the choice of marginals,the idea is to generate uniform random vectors with dependenceand then applying inverse of a distribution function to the uniform marginals based on theory (Inversion method ) to get a multivariate vector with different marginals.for the t coupla it is something like this.let's say you want to generate a vector from the t copula with dependence rho=0.5 ,matirx =[1 rho;rho 1];with 4 -degrees of freedom.after simulating the the bivariate vector apply the T_CDF with 4-degrees of freedom to the bivariate vector you simulated in the earlier step.this will generate your uniform bivariate vector ,now apply the Inversion method to the marginals with your choiceof marginal distribution taking into account the distribution parameters,you will get a bivariate vector with an imposed T copula and the choice of marginals you made.
 
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Samsaveel
Posts: 33
Joined: April 20th, 2008, 5:47 am

Simulating from Multivariate distributions or Copulas

April 3rd, 2015, 1:45 pm

the choice of copula is independent from the choice of marginals,the idea is to generate uniform random vectors with dependenceand then applying inverse of a distribution function to the uniform marginals based on theory (Inversion method ) to get a multivariate vector with different marginals.for the t coupla it is something like this.let's say you want to generate a vector from the t copula with dependence rho=0.5 ,matirx =[1 rho;rho 1];with 4 -degrees of freedom.after simulating the the bivariate vector apply the T_CDF with 4-degrees of freedom to the bivariate vector you simulated in the earlier step.this will generate your uniform bivariate vector ,now apply the Inversion method to the marginals with your choiceof marginal distribution taking into account the distribution parameters,you will get a bivariate vector with an imposed T copula and the choice of marginals you made.
 
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Tedypendah
Topic Author
Posts: 55
Joined: May 26th, 2013, 10:11 am

Simulating from Multivariate distributions or Copulas

April 3rd, 2015, 6:57 pm

Thanks Sam,I think the link to the CDF(U) is the one I was missing.......on the other hand, is there any reason 4 - degrees of freedom are popular with a T - distribution, I have another problem where I need to figure why someone used a T - distribution with 4 degrees of freedom to calculate Economic Capital.Many thanks
 
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Samsaveel
Posts: 33
Joined: April 20th, 2008, 5:47 am

Simulating from Multivariate distributions or Copulas

April 4th, 2015, 4:14 am

a t distribution with 4 DoF is in line with observed market dynamics a distribution's tail behavior
 
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Tedypendah
Topic Author
Posts: 55
Joined: May 26th, 2013, 10:11 am

Simulating from Multivariate distributions or Copulas

April 4th, 2015, 7:11 am

Hi Sam, I saw that I the Hull Options Futures and Other Derivatives book, but I thought he was just referring to the double t copula he referred to under CDO pricing.Thanks, a million. At the risk of you heckling me, any papers were we can I read up, only when you have time. Your previous paper was helpful.