G-2006-70
Efficient Correlation Matching for Normal-Copula Dependence when Univariate Marginals Are Discrete
, , and BibTeX reference
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
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