Convergence of linear approximation of Archimedean generator from Williamson's transform in examples

Rok, strany: 2017, 1 - 18

We discuss a new construction method for obtaining additive generators of Archimedean copulas proposed by McNeil, A. J.---Ne\v{s}lehov\'a, J.: \textit{Multivariate Archimedean copulas, d-monotone functions and $l_1$-norm symmetric distributions,} Ann. Statist. \textbf{37} (2009), 3059--3097, the so-called Williamson n-trans\-form, 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.

ISO 690:

Bacigál, T., Ždímalová, M. 2017. Convergence of linear approximation of Archimedean generator from Williamson's transform in examples. In Tatra Mountains Mathematical Publications, vol. 69, no.2, pp. 1-18. 1210-3195.

APA:

Bacigál, T., Ždímalová, M. (2017). Convergence of linear approximation of Archimedean generator from Williamson's transform in examples. Tatra Mountains Mathematical Publications, 69(2), 1-18. 1210-3195.