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### The Gaussian copula

The Gaussian copula is the copula that underlies the multivariate normal distribution.

 Copula name Gaussian copula Common notation Parameters , a non-negative definite matrix, i.e. a matrix that can correspond to a correlation matrix Domain Copula where is the inverse normal function and is the cumulative distribution function of the multivariate normal distribution defined by a covariance matrix equal to Kendall’s rank correlation coefficient (for bivariate case),  Where is the correlation coefficient between the two variables Coefficient of upper tail dependence,  (unless the correlation matrix exhibits perfect positive or negative dependence) Coefficient of lower tail dependence,  (unless the correlation matrix exhibits perfect positive or negative dependence) Other comments The Spearman rank correlation coefficient is given by: where is the (normal) correlation coefficient between the two variables.   If (the identity matrix) then we obtain the independence copula.

Nematrian web functions

Functions relating to the above distribution in the two dimensional case may be accessed via the Nematrian web function library by using a DistributionName of “Gaussian Copula (2d)”. For details of other supported probability distributions see here.