# Independent copula vs Student-$t$ copula with zero correlation matrix?

Suppose I have the random variables $$X_1, \dots, X_n$$ with the marginal distributions are not normal (in fact, unknown marginal distribution).

Will there be any difference between the assumption $$X_1, \dots, X_n$$ are independent and $$X_1, \dots, X_n$$ are modeled with Student-$$t$$ copula with the correlation matrix is all zeros (with $$1$$ on the diagonal)?

The uncorrelated $$t$$ copula is not the same as the independence copula. It is based on the multivariate $$t$$-distribution, which is an elliptical family, and the only elliptical distribution for which zero correlation implies independence is the normal. The difference can be quite large.

Below we will illustrate this using the R package copula. A contour plot of a $$t$$-copula is

The density of the independence copula is a constant 1. Note how the $$t$$-copula concentrates probability in the center and close the the four corners. The code used is

library(copula)
indCop <- ellipCopula(family="normal", param=0, dim=2, dispst="ex")
tCop <- ellipCopula(family="t", dim=2, dispst="ex", param=0, df=2)
getSigma(indCop)
[,1] [,2]
[1,]    1    0
[2,]    0    1
getSigma(tCop)
[,1] [,2]
[1,]    1    0
[2,]    0    1

# See they are different:
dCopula(c(0.5, 0.5), indCop)
[1] 1
dCopula(c(0.5, 0.5), tCop)
[1] 1.27324

contour(tCop, dCopula, n.grid=101, levels=c(0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.1, 1.2, 1.3), main="t copula, uncorrelated, df=2")

• Thank you very much for your answer! Could you please suggest me a paper that talks (or ideally, prove) about the fact that gaussian copula with zero correlation matrice is the same as independent copula ? Intuitively it is but rigourously, i am not sure. Thank you very much for your help! Commented Aug 6, 2021 at 21:12
• Kendall's taus = zero implies independencies only for Gaussian copulas. This is true. However, if the parameters of the t-student or Gaussian copulas are very close to its independent border and hence, they are the same as independent copulas. Just simulate a data from poor dependencies t-student copula which will be very close to independent copulas. plot(BiCopSim(500, 2, 0.02, 3)) plot(BiCopSim(500, 0,0)) Commented Aug 7, 2021 at 7:55
• So, I mean that if we fit the t-student copula to the data and estimate the parameters, which is very close to being independent, then we can assume the correlation between these variables as independent and hence fit independent copulas. However, if Kendall's tau as an independent test, shows that these variables are independent, then this is only true for Gaussian copulas. Commented Aug 7, 2021 at 8:01