Extensions of Two Bivariate Strict Archimedean Copulas

Document Type : Original Article

Author

Department of Mathematics, LMNO, University of Caen-Normandie, 14032 Caen, France

Abstract

The copula approach provides an option for capturing the structure of dependence between two quantitative variables. This approach is based on special bivariate functions called copulas. In brief, a copula can be presented as a ``functional print'' of a structure of dependence. Among the numerous types of copulas, the strict Archimedean copulas are the most famous. However, some of them appear to be unfairly neglected and understudied. We attempt to rehabilitate two of them, namely the Nelsen strict Archimedean copulas numbers 10 and 17 by extending their mathematical functionality. For each of them, the extension is made by the thorough addition of a parameter, which reveals to have a significant impact on several important aspects. As the main results, we determine the range of admissible values for the involved parameters. Then, the basic properties of the proposed extended copulas are studied, including symmetry, tail dependence, bounds, and correlation. A focus is put on the tau of Kendall as the main correlation measure. For one of the extended copulas, it is shown that there are some connections with the Frank copula and that the associated tau of Kendall almost reached the perfect interval of [-1,1], demonstrating the significance of our findings. Graphics, short statistical works, and numerical tables are given to support the theoretical results.

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Computational Journal of Mathematical and Statistical Sciences
Volume 2, Issue 2
November 2023
Pages 159-180

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