# 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
Contents

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\$. ``    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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