Constructing discrete unbounded distributions with Gaussian-copula dependence and given rank correlation

Constructing discrete unbounded distributions with Gaussian-copula dependence and given rank correlation

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(?), or the product-moment correlation, rL(?), where ? 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(?) cannot be computed exactly due to 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 ˜r(?) defined by the truncated sums, a bound on the error r(??) ? r? is given, where ?? is a solution to ˜r(??) = r?. 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.

statistics, multivariate, distribution, unbounded discrete distribution, correlation, gaussian copula

1-11

Avramidis, Athanassios N.

d6c4b6b6-c0cf-4ed1-bbe1-a539937e4001

Avramidis, Athanassios N.

d6c4b6b6-c0cf-4ed1-bbe1-a539937e4001

Avramidis, Athanassios N.
(2013)
Constructing discrete unbounded distributions with Gaussian-copula dependence and given rank correlation.
*INFORMS Journal on Computing*, n/a, .
(doi:10.1287/ijoc.2013.0563).

## 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(?), or the product-moment correlation, rL(?), where ? 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(?) cannot be computed exactly due to 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 ˜r(?) defined by the truncated sums, a bound on the error r(??) ? r? is given, where ?? is a solution to ˜r(??) = r?. 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.

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## More information

e-pub ahead of print date: 10 October 2013

Keywords:
statistics, multivariate, distribution, unbounded discrete distribution, correlation, gaussian copula

Organisations:
Operational Research

## Identifiers

Local EPrints ID: 182235

URI: https://eprints.soton.ac.uk/id/eprint/182235

ISSN: 0899-1499

PURE UUID: 96dacc81-65bb-46db-9cb9-f310f371b771

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Date deposited: 27 Apr 2011 10:21

Last modified: 18 Jul 2017 11:57

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