Efficient correlation matching for fitting discrete multivariate distributions with arbitrary marginals and normal-copula dependence

Efficient correlation matching for fitting discrete multivariate distributions with arbitrary marginals and normal-copula dependence

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 uniform 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 this root-finding problem for the case of 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, nonderivative-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

statistics, multivariate distribution, estimation, correlation, copula, simulation

88-106

Avramidis, Athanassios N.

d6c4b6b6-c0cf-4ed1-bbe1-a539937e4001

Channouf, Nabil

19ede927-0304-482c-83c5-a00376311192

L'Ecuyer, Pierre

bc8bc3bc-1eff-407b-9c57-917d045a138d

2009

Avramidis, Athanassios N.

d6c4b6b6-c0cf-4ed1-bbe1-a539937e4001

Channouf, Nabil

19ede927-0304-482c-83c5-a00376311192

L'Ecuyer, Pierre

bc8bc3bc-1eff-407b-9c57-917d045a138d

Avramidis, Athanassios N., Channouf, Nabil and L'Ecuyer, Pierre
(2009)
Efficient correlation matching for fitting discrete multivariate distributions with arbitrary marginals and normal-copula dependence.
*INFORMS Journal on Computing*, 21 (1), Winter Issue, .
(doi:10.1287/ijoc.1080.0281).

## Abstract

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 uniform 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 this root-finding problem for the case of 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, nonderivative-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

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## More information

e-pub ahead of print date: 20 August 2008

Published date: 2009

Keywords:
statistics, multivariate distribution, estimation, correlation, copula, simulation

Organisations:
Civil Maritime & Env. Eng & Sci Unit, Operational Research

## Identifiers

Local EPrints ID: 150001

URI: http://eprints.soton.ac.uk/id/eprint/150001

ISSN: 0899-1499

PURE UUID: 4017abbe-9cdf-4891-a9f9-1e0720bc0afb

## Catalogue record

Date deposited: 04 May 2010 08:27

Last modified: 14 Mar 2024 02:53

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## Contributors

Author:
Nabil Channouf

Author:
Pierre L'Ecuyer

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