Calculating correlation coefficient for Gaussian copula

Abstract: When Gaussian copula with linear correlation coefficient is used to model correlated random variables, one crucial issue is to determine a suitable correlation coefficient $\rho_z$ in normal space for two variables with correlation coefficient $\rho_x$. This paper attempts to address this problem. For two continuous variables, the marginal transformation is approximated by a weighted sum of Hermite polynomials, then, with Mehler's formula, a polynomial of $\rho_z$ is derived to approximate the function relationship between $\rho_x$ and $\rho_z$. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and $\rho_x$ is expressed as a polynomial of $\rho_z$ by Taylor expansion. For a given $\rho_x$, $\rho_z$ can be efficiently determined by solving a polynomial equation.