Matching a correlation coefficient by a Gaussian copula

ABSTRACT

A Gaussian copula is widely used to define correlated random variables. To obtain a prescribed Pearson correlation coefficient of ρ_x between two random variables with given marginal distributions, the correlation coefficient ρ_z between two standard normal variables in the copula must take a specific value which satisfies an integral equation that links ρ_x to ρ_z. 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 ρ_z is derived to approximate the function relationship between ρ_x and ρ_z. If a discrete variable is involved, the marginal transformation is decomposed into piecewise continuous ones, and ρ_x is expressed as a polynomial of ρ_z by Taylor expansion. For a given ρ_x, ρ_z can be efficiently determined by solving a polynomial equation.

KEYWORDS:

• correlation coefficient
• continuous variables
• discrete variables
• Gaussian copula
• Mehler’s formula.

MATHEMATICS SUBJECT CLASSIFICATION:

• 46N30
• 65C60
• 97K80

Acknowledgement

The authors would like to thank an anonymous reviewer for the constructive comments, which helped to improve the quality of the manuscript.

Additional information

Funding

This research is supported by National Natural Science Foundation of China (Project No. 51577057), Science and Research Fund of Hunan Province Education Department (Project No. 16C0747).