Matching a correlation coefficient by a Gaussian copula.

A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρx between two random variables with given marginal distributions, the correlation coefficient ρz between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρx to ρz. In a few cases, this equation has an explicit solution, but in other cases it must be solved numerically. This paper attempts to address this issue. If two continuous random variables are involved, the marginal transformation is approximated by a weighted sum of Hermite polynomials; via Mehler's formula, a polynomial of ρz is derived to approximate the function relationship between ρx and ρz. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρx is expressed as a polynomial of ρz by Taylor expansion. For a given ρx, ρz can be efficiently determined by solving a polynomial equation. [ABSTRACT FROM AUTHOR]