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Understanding copula transforms: a review of dependence properties

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  • Michiels F.
  • De Schepper A.

Abstract

A copula is a flexible modeling tool which contributes substantially to the study of dependencies among random variables. A broad copula class with many nice properties is the Archimedean copula class. Usually, one works with the classical bivariate models, e.g. as summarized in Nelsen (2006), which are one-parametric models. However, in many cases when practitioners want to model dependencies by means of copulas, it would be more rational to work with multi-parametric models. Indeed, multi-parametric models would allow to better harmonize empirical information with the model, as it would be possible to directly import more than one characteristic into the model, e.g. measures of concordance, tail dependence and so on. Various ways exist and have been explored to define multi-parameter Archimedean models. This paper intends to elaborate on one particular method, namely the technique of transforms. More specifically, the contribution of this article is threefold: 1. Genest et al. (1998) sum up five feasible transformations applicable on the Archimedean generator '. In this note we present an overview of these transformations by generalizing tail dependence properties and limiting cases. 2. In an earlier paper, see Michiels et al. (2008), we showed that it can be advantageous to work with the -function instead of with the generator function. We investigate here the effect of transforms on this -function. 3. We introduce a new type of transform which is concordance invariant. As such, this type of transform has practical use as it allows to create comparable test spaces (see Michiels and De Schepper (2008)) from a particular copula family. The paper is organised as follows. In section two the most important copula properties are discussed, with the focus on the Archimedean class. Next, section three reviews generally known copula transforms and generator transforms. In section four the effect of the transforms on the -level is being discussed, which allows the derivation of general tail dependence properties and limiting cases. We also introduce a concordance invariant transform and illustrate its use through simulations. Finally, section five concludes.

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Bibliographic Info

Paper provided by University of Antwerp, Faculty of Applied Economics in its series Working Papers with number 2009012.

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Length: 23 pages
Date of creation: Dec 2009
Date of revision:
Handle: RePEc:ant:wpaper:2009012

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Web page: https://www.uantwerp.be/en/faculties/applied-economic-sciences/
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Cited by:
  1. Frederik Michiels & Ann De Schepper, 2012. "How to improve the fit of Archimedean copulas by means of transforms," Statistical Papers, Springer, vol. 53(2), pages 345-355, May.

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