<abstract xmlns="http://www.w3.org/1999/xhtml"><p> We discuss a new construction method for obtaining additive generators of Archimedean copulas proposed by McNeil, A. J.-Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and l<sub>1</sub>-norm symmetric distributions, Ann. Statist. 37 (2009), 3059-3097, the so-called Williamson n-transform, and illustrate it by several examples. We show that due to the equivalence of convergences of positive distance functions, additive generators and copulas, we may approximate any n-dimensional Archimedean copula by an Archimedean copula generated by a transformation of weighted sum of Dirac functions concentrated in certain suitable points. Specifically, in two dimensional case this means that any Archimedean copula can be approximated by a piece-wise linear Archimedean copula, moreover the approximation of generator by linear splines circumvents the problem with the non-existence of explicit inverse.</p></abstract>

We discuss a new construction method for obtaining additive generators of Archimedean copulas proposed by McNeil, A. J.-Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and l1-norm symmetric distributions, Ann. Statist. 37 (2009), 3059-3097, the so-called Williamson n-transform, and illustrate it by several examples. We show that due to the equivalence of convergences of positive distance functions, additive generators and copulas, we may approximate any n-dimensional Archimedean copula by an Archimedean copula generated by a transformation of weighted sum of Dirac functions concentrated in certain suitable points. Specifically, in two dimensional case this means that any Archimedean copula can be approximated by a piece-wise linear Archimedean copula, moreover the approximation of generator by linear splines circumvents the problem with the non-existence of explicit inverse.

Convergence of Linear Approximation of Archimedean Generator from Williamson’s Transform in Examples

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

We discuss a new construction method for obtaining additive generators of Archimedean copulas proposed by McNeil, A. J.-Nešlehová, J.: Multivariate Archimedean...

Convergence of Linear Approximation of Archimedean Generator from Williamson’s Transform in Examples

Published Online: Mar 23, 2018

Page range: 1 - 18

Received: Apr 13, 2017

DOI: https://doi.org/10.1515/tmmp-2017-0010

Keywords
Archimedean copula, Williamson, approximation

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Convergence of Linear Approximation of Archimedean Generator from Williamson’s Transform in Examples

Published Online: Mar 23, 2018

Page range: 1 - 18

Received: Apr 13, 2017

DOI: https://doi.org/10.1515/tmmp-2017-0010

KeywordsArchimedean copula, Williamson, approximation

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Archimedean copula, Williamson, approximation