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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
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
0899-1499
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
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, 88-106. (doi:10.1287/ijoc.1080.0281).

Record type: Article

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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e-pub ahead of print date: 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: https://eprints.soton.ac.uk/id/eprint/150001
ISSN: 0899-1499
PURE UUID: 4017abbe-9cdf-4891-a9f9-1e0720bc0afb
ORCID for Athanassios N. Avramidis: ORCID iD orcid.org/0000-0001-9310-8894

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Date deposited: 04 May 2010 08:27
Last modified: 05 Nov 2019 01:45

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Contributors

Author: Nabil Channouf
Author: Pierre L'Ecuyer

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