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

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
0899-1499
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, 1-11. (doi:10.1287/ijoc.2013.0563).

Record type: Article

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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e-pub ahead of print date: 10 October 2013
Keywords: statistics, multivariate, distribution, unbounded discrete distribution, correlation, gaussian copula
Organisations: Operational Research

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Local EPrints ID: 182235
URI: https://eprints.soton.ac.uk/id/eprint/182235
ISSN: 0899-1499
PURE UUID: 96dacc81-65bb-46db-9cb9-f310f371b771
ORCID for Athanassios N. Avramidis: ORCID iD orcid.org/0000-0001-9310-8894

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Date deposited: 27 Apr 2011 10:21
Last modified: 24 Jul 2019 00:34

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