Bivariate copulas play a central role in modeling the dependence structure between two random variables and serve as a fundamental tool in various applied fields. In this article, we develop a new theoretical framework aimed at constructing simple asymmetric bivariate copulas of the form $ C(u, v) = u v \left[\phi(v) + u (1-\phi(v))\right] $, $ (u, v)\in [0, 1]^2 $. This framework relies on a tuning univariate function to achieve the desired asymmetry. We study this pioneering scheme, emphasizing its theoretical foundations, and illustrating it with several examples. More precisely, we establish important properties of the proposed copulas and derive analytical expressions for concordance measures such as Spearman's rho, Kendall's tau, Gini's gamma, and Blomqvist's beta. In addition, we investigate the estimation procedure for the dependence parameter using the maximum likelihood approach. Finally, we conduct a simulation study to evaluate the performance of the proposed estimator. A real climatological dataset from the city of Abu Dhabi is used to demonstrate the applicability of the proposed copulas, with very convincing results.
Citation: Rachid Bentoumi, Farid El Ktaibi, Christophe Chesneau. A new scheme for simple asymmetric bivariate copulas and applications[J]. AIMS Mathematics, 2025, 10(10): 24602-24626. doi: 10.3934/math.20251091
Bivariate copulas play a central role in modeling the dependence structure between two random variables and serve as a fundamental tool in various applied fields. In this article, we develop a new theoretical framework aimed at constructing simple asymmetric bivariate copulas of the form $ C(u, v) = u v \left[\phi(v) + u (1-\phi(v))\right] $, $ (u, v)\in [0, 1]^2 $. This framework relies on a tuning univariate function to achieve the desired asymmetry. We study this pioneering scheme, emphasizing its theoretical foundations, and illustrating it with several examples. More precisely, we establish important properties of the proposed copulas and derive analytical expressions for concordance measures such as Spearman's rho, Kendall's tau, Gini's gamma, and Blomqvist's beta. In addition, we investigate the estimation procedure for the dependence parameter using the maximum likelihood approach. Finally, we conduct a simulation study to evaluate the performance of the proposed estimator. A real climatological dataset from the city of Abu Dhabi is used to demonstrate the applicability of the proposed copulas, with very convincing results.
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Rachid Bentoumi, Farid El Ktaibi, Christophe Chesneau. A new scheme for simple asymmetric bivariate copulas and applications[J]. AIMS Mathematics, 2025, 10(10): 24602-24626. doi: 10.3934/math.20251091
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