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I want to simulate a t copula with given correlation parameter $\Sigma$ and $k$ degrees of freedom. I can't find any literature about practical simulation, so I am trying new approaches. (I also have some problems using packages in R. I am working on that but I feel like I can find a faster and more interesting solution to my problem)

A student r.v. with k degree of freedom has the same distribution as $Z\sqrt{\frac{k}{V}}$ where $Z$ follow a normal with mean 0 and variance 1 and $V$ follow a chi-squared distribution with $k$ degrees of freedom.

If I simulate a multivariate (n dimension) normal r.v. with mean 0 and variance $\Sigma$ and divide them by n independant rv following a chi-squared distibution I can obtain a multivariate r.v. where each one follow a t ditribution and the correlation is $\Sigma$. Am I right ? Is this equivalent to simulating a n-dimensionnal t-student r.v. with parameter k and correlation $\Sigma$ ?

Then I want to "go back" to quantiles. With a multivariate normal r.v. I would use the inverse cdf on each r.v. Does this hold with a t-distibution ? If I use the inverse cdf of a univariate t-distribution on each of my n r.v. that follow a t distribution will I get the same result as simulating a t-copula with parameters $\Sigma$ and k ?

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The correct way to obtain a simulation from a multivariate t-distribution is to simulate from a correlated multivariate normal and then divide each component by the same denominator.

![enter image description here

If you then convert each component individually to a uniform, you have a sample from a t-copula.

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  • $\begingroup$ when you say converting, is it with the inverse cdf of student-t ? $\endgroup$ – lcrmorin Nov 5 '15 at 8:16
  • $\begingroup$ @Were_cat No, with the cdf. If a random variable $X$ has cdf $F_X$ then $U\sim F_X(X)$ is distributed as uniform. $\endgroup$ – Glen_b Nov 5 '15 at 9:05
  • $\begingroup$ yes, of course I meant the c.d.f. of student-t. Thanks a lot $\endgroup$ – lcrmorin Nov 5 '15 at 10:24

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