Solved – Relation between Spearman and Pearson correlation in Gaussian copula

Is one of these statements correct for Gauss copula?

  • For normal marginals Pearson correlation equal to Spearman correlation
  • Pearson correlation is less or equal than Spearman correlation

A dependence measure is a parameter of a Gauss copula. Let denote by $rho$ the Pearson correlation, $rho_s$ the Spearman correlation, and $tau_k$ the Kendell tau.

It's known $rho in [-1,1]$, and $$rho_s = frac{6}{pi}cdot arcsin(frac{rho}{2}) ,$$ $$tau_k = frac{2}{pi}cdot arcsin(rho).$$

In the plot below one can see the dependencies between $rho_s$, $tau_k$ and $rho$.

enter image description here

    rho <- seq(-1,1,0.01)     # Pearson correlation     rho_s <- 6/pi*asin(rho/2) # Spearman correlation     tau_k <- 2/pi*asin(rho)   # Kendell tau     plot(rho, rho, type="n", xlab="rho", ylab="rho_s, tau_k")     lines(rho, tau_k, col = "red")     lines(rho, rho_s, col = "blue")     legend("topleft", lty=c(1,1), col = c("red", "blue"), legend=c("tau_k", "rho_s")) 

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