Investigation of a variable-power FGM-type copula

Document Type : Original Article

Author

• Christophe Chesneau

Department of Mathematics, University of Caen Normandie, France

10.22067/smps.2024.45178

Abstract

Copula theory is essentially based on the use of multivariate functions, commonly called copulas. These functions provide a versatile toolset for capturing a broad spectrum of dependence structures, highlighting their indispensability in a multitude of applied domains. However, in light of the evolving complexities inherent in real-world data, there is a growing demand for pioneering copula constructions. In this paper, our main goal is to increase the field of variable-power copulas by introducing an innovative Farlie-Gumbel-Morgenstern (FGM)-type power copula. It is distinguished by a unique one-parameter formulation that recovers the independence copula. In the main part, we establish its mathematical validity, which is based on differentiation techniques, appropriate factorizations, and two complementary logarithmic inequalities. Then we provide a comprehensive exploration of its modeling properties, with a focus on its negative dependence through the beta medial correlation, rho of Spearman and tau of Kendall. The corresponding copula data generation is examined with different values of the parameter. A new bivariate normal distribution is also derived, and its shapes are discussed. Finally, the minimum and maximum of two random variables connected through the proposed copula are examined from a distributional viewpoint. Our findings contribute to the advancement of copula theory, thereby enhancing its practical utility across a wide range of disciplines.

Keywords

• ‎Bivariate distributions
• Bivariate plots
• Copula approach
• Correlation measures
• Data generation
• ‎ Dependence model
• Inequalities

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