# Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

Alireza Daneshkhah, G Parham, O Chatrabgoun, M Jokar

Research output: Contribution to journalArticle

6 Citations (Scopus)

### Abstract

We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n − 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.

The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.
Original language English 389-419 31 Communications in Statistics - Simulation and Computation 45 2 5 Nov 2015 https://doi.org/10.1080/03610918.2013.804557 Published - 2016

### Fingerprint

Orthonormal Polynomials
Multiwavelet
Multivariate Distribution
Copula
Legendre
Basis Functions
Polynomials
Approximation
Series
Expert Judgment
Copula Models
Semiparametric Methods
Financial Data
Stacking
Graphical Models
Approximation Methods
n-dimensional
High-dimensional
Polynomial
Modeling

### Keywords

• Pair-copula Construction
• Density estimation

### Cite this

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title = "Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions",
abstract = "We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n − 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.",
keywords = "Pair-copula Construction, Density estimation",
author = "Alireza Daneshkhah and G Parham and O Chatrabgoun and M Jokar",
year = "2016",
doi = "10.1080/03610918.2013.804557",
language = "English",
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pages = "389--419",
journal = "Communications in Statistics Part B: Simulation and Computation",
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T1 - Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

AU - Daneshkhah, Alireza

AU - Parham, G

AU - Chatrabgoun, O

AU - Jokar, M

PY - 2016

Y1 - 2016

N2 - We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n − 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.

AB - We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n − 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.

KW - Pair-copula Construction

KW - Density estimation

U2 - 10.1080/03610918.2013.804557

DO - 10.1080/03610918.2013.804557

M3 - Article

VL - 45

SP - 389

EP - 419

JO - Communications in Statistics Part B: Simulation and Computation

JF - Communications in Statistics Part B: Simulation and Computation

SN - 0361-0918

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ER -