Group for Research in Decision Analysis

# 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 $\rho$ 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 $\rho$. 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 $\rho$. 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 $\rho$) to the continuous-marginals limiting function, when the discrete marginals converge to continuous distributions.

, 38 pages