### Abstract

Apopular 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 r 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 p. 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 p. 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-theart, nonderivative-based method. We also characterize the asymptotic convergence rate of the function r (as a function of p) to the continuous-marginals limiting function, when the discrete marginals converge to continuous distributions.

Original language | English |
---|---|

Pages (from-to) | 88-106 |

Number of pages | 19 |

Journal | INFORMS Journal on Computing |

Volume | 21 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2009 |

### Fingerprint

### Keywords

- Copula
- Correlation
- Estimation
- Multivariate distribution
- Simulation
- Statistics

### ASJC Scopus subject areas

- Software
- Information Systems
- Computer Science Applications
- Management Science and Operations Research

### Cite this

*INFORMS Journal on Computing*,

*21*(1), 88-106. https://doi.org/10.1287/ijoc.1080.0281

**Efficient correlation matching for fitting discrete multivariate distributions with arbitrary marginals and normal-copula dependence.** / Avramidis, Athanassios N.; Channouf, Nabil; L'Ecuyer, Pierre.

Research output: Contribution to journal › Article

*INFORMS Journal on Computing*, vol. 21, no. 1, pp. 88-106. https://doi.org/10.1287/ijoc.1080.0281

}

TY - JOUR

T1 - Efficient correlation matching for fitting discrete multivariate distributions with arbitrary marginals and normal-copula dependence

AU - Avramidis, Athanassios N.

AU - Channouf, Nabil

AU - L'Ecuyer, Pierre

PY - 2009/1

Y1 - 2009/1

N2 - Apopular 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 r 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 p. 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 p. 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-theart, nonderivative-based method. We also characterize the asymptotic convergence rate of the function r (as a function of p) to the continuous-marginals limiting function, when the discrete marginals converge to continuous distributions.

AB - Apopular 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 r 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 p. 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 p. 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-theart, nonderivative-based method. We also characterize the asymptotic convergence rate of the function r (as a function of p) to the continuous-marginals limiting function, when the discrete marginals converge to continuous distributions.

KW - Copula

KW - Correlation

KW - Estimation

KW - Multivariate distribution

KW - Simulation

KW - Statistics

UR - http://www.scopus.com/inward/record.url?scp=67649977091&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67649977091&partnerID=8YFLogxK

U2 - 10.1287/ijoc.1080.0281

DO - 10.1287/ijoc.1080.0281

M3 - Article

VL - 21

SP - 88

EP - 106

JO - INFORMS Journal on Computing

JF - INFORMS Journal on Computing

SN - 1091-9856

IS - 1

ER -