Constructing Discrete Unbounded Distributions with Gaussian-Copula Dependence and Given Rank Correlation

Abstract

A random vector X with given univariate marginals can be obtained by first applying the normal distribution function to each coordinate of a vector Z of correlated standard normals to produce a vector U of correlated uniforms over (0,1) and then transforming each coordinate of U by the relevant inverse marginal. One approach to fitting requires, separately for each pair of coordinates of X, the rank correlation, r(ρ), or the product-moment correlation, rL(ρ), where ρ is the correlation of the corresponding coordinates of Z, to equal some target r*. We prove the existence and uniqueness of a solution for any feasible target, without imposing restrictions on the marginals. For the case where r(ρ) cannot be computed exactly because of an infinite discrete support, the relevant infinite sums are approximated by truncation, and lower and upper bounds on the truncation errors are developed. With a function [Formula: see text] defined by the truncated sums, a bound on the error r(ρ*) − r* is given, where ρ* is a solution to [Formula: see text]. Based on this bound, an algorithm is proposed that determines truncation points so that the solution has any specified accuracy. The new truncation method has potential for significant work reduction relative to truncating heuristically, largely because as required accuracy decreases, so does the number of terms in the truncated sums. This is quantified with examples. The gain appears to increase with the heaviness of tails.