### G-2006-70

# Efficient Correlation Matching for Normal-Copula Dependence when Univariate Marginals Are Discrete

## Athanassios N. Avramidis, Nabil Channouf, and Pierre L'Ecuyer

A popular approach for modeling dependence in a finite-dimensional random vector
**X** with given univariate marginals is via a normal copula that fits the rank or linear
correlations for the bivariate marginals of **X**. In this approach, known as the NORTA
method, the normal distribution function is applied to each coordinate of a vector **Z**
of correlated standard normals to produce a vector **U** of correlated uniforms random
variables over (0, 1); then **X** is obtained by applying the inverse of the target marginal
distribution function for each coordinate of **U**. The fitting requires finding the appropriate
correlation between any two given coordinates of **Z** that would yield the
target rank or linear correlation *r* between the corresponding coordinates of **X**. This
root-finding problem is easy to solve when the marginals are continuous, but not when
they are discrete. In this paper, we provide a detailed analysis of the NORTA method
for discrete marginals. We prove key properties of r and of its derivative as a function
of . It turns out that the derivative is easier to evaluate than the function itself.
Based on that, we propose and compare alternative methods for finding or approximating
the appropriate . The case of discrete distributions with unbounded support
is covered as well. In our numerical experiments, a derivative-supported method is
faster and more accurate than a state-of-the-art, non-derivative-based method. We
also characterize the asymptotic convergence rate of the function *r* (as a function of )
to the continuous-marginals limiting function, when the discrete marginals converge
to continuous distributions.

Published **November 2006**
,
38 pages