How to improve the fit of Archimedean copulas by means of transforms

The selection of copulas is an important aspect of dependence modeling issues. In many practical applications, only a limited number of copulas is tested and the copula with the best result for a goodness-of-fit test is chosen, which, however, does not always lead to the best possible fit. In this paper we develop a practical and logical method for improving the goodness-of-fit of a particular Archimedean copula by means of transforms. In order to do this, we introduce concordance invariant transforms which can also be tail dependence preserving, based on an analysis on the λ-function, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lambda=\frac{\varphi}{\varphi'}}$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varphi}$$\end{document} is the Archimedean generator. The methodology is applied to the data set studied in Cook and Johnson (J R Stat Soc B 43:210–218, 1981) and Genest and Rivest (J Am Stat Assoc 88:1043–1043, 1993), where we improve the fit of the Frank copula and obtain statistically significant results.