For elliptical copulae (Gaussian, t-student) there are known relationships between the linear correlation (Pearson), rho, and the rank correlations Spearman’s rho_s and Kendall’s tau. The relationships are given by

rho = sin(tau pi/2)

rho = 2 sin(rho_s pi/6)

Below a small table with values

When sampling a Gaussian or t-Student copula, one needs to specify a linear correlation matrix containing Pearson correlation coefficients.

I played around with the copulae Python package and did some 2D simulations, and computed the Spearman’s rho and Kendall’s tau. The computed rank coefficients match very well to the analytical expressions above when I apply this to the Gaussian copula.

However, when doing the same applied to the t-Student distribution with 1 degree of freedom, I get similar values for tau as in the above table, but not for Spearman’s rho. I was under the impression that the relationship between rho and rho_s holds for both the Gaussian and the t-Student copula, given that they are both rank correlation coefficients that are invariant under monotonic transformations.

However, I get different results as can be seen in the table below, under the header ‘t-Student’

My question is: is it indeed true that the relationships above hold for both Gaussian and t-Student distributions? And if so, why do I get almost the same tau value between Gaussian and t-Student copula for a given rho, but not for the Spearman’s rho_s?