Constructions of absolutely continuous copulas with given curvilinear or opposite-curvilinear sections

Document Type : Research Paper

Authors

1 School of Science, Nanchang Institute of Technology

2 School of Business Administration, Nanchang Institute of Technology, China

Abstract

In this paper, we address the constructions of absolutely continuous copulas characterized by given
curvilinear or opposite-curvilinear sections. For a given function generated by a suitable curve within the unit square, we develop a family of absolutely continuous copulas having this function as a curvilinear section, achieved by averaging over a series of copulas sharing the common curvilinear section. Subsequently, we establish theoretical relationships between absolutely continuous copulas with given opposite-curvilinear sections and those with corresponding curvilinear sections. Using these relationships, we propose two approaches for constructing of absolutely continuous copulas with given opposite-curvilinear sections.

Keywords

Main Subjects

References

[1] A. E. Abbas, Multiattribute utility copulas, Operations Research, 57(6) (2009), 1367-1383. https://doi.org/10.
1287/opre.1080.0687
[2] D. K. Bukovˇsek, B. Mojˇskerc, N. Stopar, Exact upper bound for copulas with a given diagonal section, Fuzzy Sets
and Systems, 480 (2024), 108865. https://doi.org/10.1016/j.fss.2024.108865
[3] C. Butucea, J. F. Delmas, A. Dutfoy, R. Fischer, Maximum entropy copula with given diagonal section, Journal of
Multivariate Analysis, 137 (2015), 61-81. https://doi.org/10.1016/j.jmva.2015.01.003
[4] B. De Baets, H. De Meyer, T. Jwaid, On the degree of asymmetry of a quasi-copula with respect to a curve, Fuzzy
Sets and Systems, 354(1) (2019), 84-103. https://doi.org/10.1016/j.fss.2018.05.002
[5] F. Durante, J. A. Rodr´ıguez-Lallena, M. ´Ubeda-Flores, New constructions of diagonal patchwork copulas, Information
Sciences, 179(19) (2009), 3383-3391. https://doi.org/10.1016/j.ins.2009.06.007
[6] F. Durante, C. Sempi, Principles of copula theory, Chapman and Hall/CRC, Boca Raton, 2016. https://doi.org/
10.1201/b18674
[7] J. Fern´andez-S´anchez, M. ´Ubeda-Flores, Copulas with given track and opposite track sections: Solution to a problem
on diagonals, Fuzzy Sets and Systems, 308(1) (2017), 133-137. https://doi.org/10.1016/j.fss.2016.06.011
[8] G. A. Fredricks, R. B. Nelsen, The Bertino family of copulas, In: C. M. Cuadras, J. Fortiana, J. A. Rodriguez-
Lallena (Eds.), Distributions with given marginals and statistical modelling, Springer, Dordrecht, 2002. https:
//doi.org/10.1007/978-94-017-0061-0_10
[9] P. H´ajek, R. Mesiar, On copulas, quasicopulas and fuzzy logic, Soft Computing, 12(12) (2008), 1239-1243. https:
//doi.org/10.1007/s00500-008-0286-z
[10] H. Joe, Multivariate models and multivariate dependence concepts, Chapman and Hall, London, 1997. https:
//doi.org/10.1201/9780367803896
[11] T. Jwaid, H. De Meyer, A. Haj Ismail, B. De Baets, Curved splicing of copulas, Information Sciences, 556 (2021),
95-110. https://doi.org/10.1016/j.ins.2020.12.053
[12] E. P. Klement, A. Koles´arov´a, R. Mesiar, C. Sempi, Copulas constructed from horizontal sections, Communications
in Statistics - Theory and Methods, 36(16) (2007), 2901-2911. https://doi.org/10.1080/03610920701386976
[13] Q. Lou, H. M. Zhang, Y. S. Ye, Curved splicing constructions of (quasi-)copulas with given opposite track sections,
Iranian Journal of Fuzzy Systems, 21(4) (2024), 81-100. https://doi.org/10.22111/ijfs.2024.48824.8612
[14] R. B. Nelsen, An introduction to copulas, Second Edition, Springer, New York, 2006. https://doi.org/10.1007/
0-387-28678-0
[15] R. B. Nelsen, J. J. Quesada-Molina, J. A. Rodr´ıguez-Lallena, M. ´Ubeda-Flores, Best-possible bounds on sets of
bivariate distribution functions, Journal of Multivariate Analysis, 90(2) (2004), 348-358. https://doi.org/10. 1016/j.jmva.2003.09.002
[16] R. B. Nelsen, J. J. Quesada-Molina, J. A. Rodr´ıguez-Lallena, M. ´Ubeda-Flores, On the construction of copulas
and quasi-copulas with given diagonal sections, Insurance: Mathematics and Economics, 42(2) (2008), 473-483.
https://doi.org/10.1016/j.insmatheco.2006.11.011
[17] Y. Ouyang, Y. H. Sun, H. P. Zhang, On an upper bound of the set of copulas with a given curvilinear section,
Fuzzy Sets and Systems, 500 (2025), 109199. https://doi.org/10.1016/j.fss.2024.109199
[18] A. Sancetta, S. Satchell, The Bernstein copula and its applications to modeling and approximations of multivariate
distributions, Econometric Theory, 20(3) (2004), 535-562. https://doi.org/10.1017/S026646660420305X
[19] A. Sklar. Fonctions de r´epartition `a n dimensions et leurs marges, Publications de l’Institut de Statistique de
l’Universit´e Paris, 8 (1959), 229-231. http://doi.org/10.2139/ssrn.4198458
[20] M. ´Ubeda-Flores, On the best-possible upper bound on sets of copulas with given diagonal sections, Soft Computing, 12(10) (2008), 1019-1025. https://doi.org/10.1007/s00500-007-0269-5
[21] J. H. Xie, J. Fang, J. P. Yang, L. Bu, Multivariate composite copulas, Astin Bulletin, 52(1) (2022), 145-184.
https://doi.org/10.1017/asb.2021.30
[22] J. H. Xie, B. Y. Wu, W. Zou, C. Y. Jiang, Curvilinear patchwork constructions of (quasi-) copulas with given
curvilinear sections, Fuzzy Sets and Systems, 473 (2023), 108720. https://doi.org/10.1016/j.fss.2023.108720
[23] W. H. Zhu, L. J. Li, J. P. Yang, J. H. Xie, L. L. Sun, Asymptotic subadditivity/superadditivity of Value-at-Risk
under tail dependence, Mathematical Finance, 33(4) (2023), 1314-1369. https://doi.org/10.1111/mafi.12393
[24] W. Zou, L. L. Sun, J. H. Xie, Best-possible bounds on the sets of copulas and quasi-copulas with given curvilinear
sections, Fuzzy Sets and Systems, 441 (2022), 335-365. https://doi.org/10.1016/j.fss.2021.12.008
Iranian Journal of Fuzzy Systems
Volume 22, Issue 5
September and October 2025
Pages 111-128

Files

Share

How to cite

Statistics