Copula with a certain correlation

Question

What does it mean for values to be "drawn from a normal copula with correlation $\rho\in [0, 1]$"? Is that a normal distribution with a covariance matrix whose entries are uniform random in $[0, 1]$?

Would sampling from a multivariate normal distribution using a probability package (like numpy) be equivalent to "drawn from a normal copula with correlation $\rho\in [0, 1]$"?

Answer 1

It certainly does not mean what you suggest. The number $\rho\in[0,1]$ is not random.

It means you have two random variables $U,V,$ each marginally uniformly distributed, so that $X = \Phi_{\mu,\sigma^2}^{-1}(U)$ and $Y=\Phi_{\nu,\tau^2}^{-1}(V)$ are normally distributed, and and the joint distribution of $U,V$ is such that

• $\operatorname{corr}(X,Y)=\rho,$ and
• Every constant (i.e. non-random) linear combination of $X$ and $Y$ is normally distributed.

You can also do this with more than two random variables, in which case the correlation between any two of them is $\rho.$

The joint distribution of $(U,V)$ is the copula. Generally, a copula is a multivariate distribution whose marginals are $\operatorname{Uniform}(0,1).$