Document Type : Research Paper
Authors
School of Mathematics and Statistics, Guilin University of Technology, Guilin, China
Abstract
In this paper, our focus is to construct (quasi-)copulas with a given opposite track section. Three different ways for constructing (quasi-)copulas with a given opposite track section are proposed and the best-possible bounds for all those (quasi-)copulas sharing a common opposite track section are also discussed. Drawing inspiration from the opposite diagonal splicing operation, we propose the curved splicing operation. We obtain that the curved splicing of any two quasi-copulas with a common opposite track section must be a quasi-copula. We also show several sufficient conditions
for the curved splicing of any two copulas sharing a common opposite track section to be a copula. Furthermore, we discuss the characteristics of a copula constructed by the curved splicing operation. Finally, we construct two novel types of semilinear copulas, and by applying the curved splicing operation we obtain two additional types of semilinear copulas.
Keywords
Main Subjects
[1] A. A. Al-Shomrani, New bivariate family of distributions based on any copula function: Statistical properties,
Heliyon, 9 (2023), e15160. https://doi.org/10.1016/j.heliyon.2023.e15160
Heliyon, 9 (2023), e15160. https://doi.org/10.1016/j.heliyon.2023.e15160
[2] C. Alsina, R. Nelsen, B. Schweizer, On the characterization of a class of binary operations on distribution functions,
Statistics and Probability Letters, 17(2) (1993), 85-89. https://doi.org/10.1016/0167-7152(93)90001-Y
Statistics and Probability Letters, 17(2) (1993), 85-89. https://doi.org/10.1016/0167-7152(93)90001-Y
[3] C. Amblard, S. Girard, A new extension of bivariate FGM copulas, Metrika, 70 (2009), 1-17. https://doi.org/
10.1007/s00184-008-0174-7
10.1007/s00184-008-0174-7
[4] J. J. Arias-Garc´ıa, R. Mesiar, B. De Baets, A hitchhiker’s guide to quasi-copulas, Fuzzy Sets and Systems, 393
(2020), 1-28. https://doi.org/10.1016/j.fss.2019.06.009
(2020), 1-28. https://doi.org/10.1016/j.fss.2019.06.009
[5] 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
Multivariate Analysis, 137 (2015), 61-81. https://doi.org/10.1016/j.jmva.2015.01.003
[6] C. M. Cuadras, J. L. Aug´e, A continuous general multivariate distribution and its properties, Communications in
Statistics-Theory and Methods, 10 (1981), 339-353. https://doi.org/10.1080/03610928108828042
Statistics-Theory and Methods, 10 (1981), 339-353. https://doi.org/10.1080/03610928108828042
[7] C. M. Cuadras, J. Fortiana, J. A. Rodr´ıguez-Lallena, Distributions with given marginals and statistical modelling,
Springer Netherlands, 2002. https://doi.org/10.1007/978-94-017-0061-0
Springer Netherlands, 2002. https://doi.org/10.1007/978-94-017-0061-0
[8] E. De Amo, H. De Meyer, M. D´ıaz Carrillo, J. Fern´andez-S´anchez, Characterization of copulas with given diagonal
and opposite diagonal sections, Fuzzy Sets and Systems, 284 (2016), 63-77. https://doi.org/10.1016/j.fss.
2014.10.030
and opposite diagonal sections, Fuzzy Sets and Systems, 284 (2016), 63-77. https://doi.org/10.1016/j.fss.
2014.10.030
[9] 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 (2019), 84-103. https://doi.org/10.1016/j.fss.2018.05.002
Sets and Systems, 354 (2019), 84-103. https://doi.org/10.1016/j.fss.2018.05.002
[10] B. De Baets, H. De Meyer, J. Kalick´a, R. Mesiar, Flipping and cyclic shifting of binary aggregation functions,
Fuzzy Sets and Systems, 160 (2009), 752-765. https://doi.org/10.1016/j.fss.2008.03.008
Fuzzy Sets and Systems, 160 (2009), 752-765. https://doi.org/10.1016/j.fss.2008.03.008
[11] B. De Baets, H. De Meyer, J. Kalick´a, R. Mesiar, On the relationship between modular functions and copulas,
Fuzzy Sets and Systems, 268 (2015), 110-126. https://doi.org/10.1016/j.fss.2014.07.024
Fuzzy Sets and Systems, 268 (2015), 110-126. https://doi.org/10.1016/j.fss.2014.07.024
[12] B. De Baets, H. De Meyer, R. Mesiar, Asymmetric semilinear copulas, Kybernetika, 43 (2007), 221-233. https:
//www.kybernetika.cz/content/2007/2/221/paper.pdf
//www.kybernetika.cz/content/2007/2/221/paper.pdf
[13] B. De Baets, H. De Meyer, M. ´Ubeda-Flores, Opposite diagonal sections of quasi-copulas and copulas, International
Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17 (2009), 481-490. https://doi.org/10.1142/
S0218488509006108
Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17 (2009), 481-490. https://doi.org/10.1142/
S0218488509006108
[14] M. Denuit, J. Dhaene, G. Marc, R. Kaas, Actuarial theory for dependent risks: Measures, orders and models, John
Wiley and Sons, Ltd, 2005. https://doi.org/10.1002/0470016450
Wiley and Sons, Ltd, 2005. https://doi.org/10.1002/0470016450
[15] F. Durante, J. Fern´andez-S´anchez, R. Pappad´a, Copulas, diagonals, and tail dependence, Fuzzy Sets and Systems,
264 (2015), 22-41. https://doi.org/10.1016/j.fss.2014.03.014
264 (2015), 22-41. https://doi.org/10.1016/j.fss.2014.03.014
[16] F. Durante, A. Koles´arov´a, R. Mesiar, C. Sempi, Semilinear copulas, Fuzzy Sets and Systems, 159 (2008), 63-76.
https://doi.org/10.1016/j.fss.2007.09.001
https://doi.org/10.1016/j.fss.2007.09.001
[17] F. Durante, R. Mesiar, P. L. Papini, C. Sempi, 2-Increasing binary aggregation operators, Information Sciences,
177(1) (2007), 111-129. https://doi.org/10.1016/j.ins.2006.04.006
177(1) (2007), 111-129. https://doi.org/10.1016/j.ins.2006.04.006
[18] F. Durante, J. A. Rodr´ıguez-Lallena, M. ´Ubeda-Flores, New constructions of diagonal patchwork copulas, Information
Science, 179 (2009), 3383-3391. https://doi.org/10.1016/j.ins.2009.06.007
Science, 179 (2009), 3383-3391. https://doi.org/10.1016/j.ins.2009.06.007
[19] P. Embrechts, A. H¨oing, A. Juri, Using copulae to bound the Value-at-Risk for functions of dependent risks, Finance
and Stochastics, 7 (2003), 145-167. https://doi.org/10.1007/s007800200085
and Stochastics, 7 (2003), 145-167. https://doi.org/10.1007/s007800200085
[20] M. Esfahani, M. Amini, G. R. Mohtashami-Borzadaran, A. Dolati, A new copula-based bivariate Gompertz–
Makeham model and its application to COVID-19 mortality data, Iranian Journal of Fuzzy Systems, 20(3) (2023),
159-175. https://doi.org/10.22111/IJFS.2023.7645
Makeham model and its application to COVID-19 mortality data, Iranian Journal of Fuzzy Systems, 20(3) (2023),
159-175. https://doi.org/10.22111/IJFS.2023.7645
[21] 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 (2017), 133-137. https://doi.org/10.1016/j.fss.2016.06.011
[22] H. Joe, Multivariate models and multivariate dependence concepts, Taylor and Francis, 1997. https://doi.org/
10.1201/9780367803896
10.1201/9780367803896
[23] T. Jwaid, Semilinear and semiquadratic conjunctive aggregation functions, PhD Thesis, Ghent University, 2014.
https://biblio.ugent.be/publication/5702047
https://biblio.ugent.be/publication/5702047
[24] T. Jwaid, B. De Baets, H. De Meyer, Biconic aggregation functions, Information Sciences, 187 (2012), 129-150.
https://doi.org/10.1016/j.ins.2011.10.012
https://doi.org/10.1016/j.ins.2011.10.012
[25] T. Jwaid, B. De Baets, H. De Meyer, Ortholinear and paralinear semi-copulas, Fuzzy Sets and Systems, 252 (2014),
76-98. https://doi.org/10.1016/j.fss.2014.02.004
76-98. https://doi.org/10.1016/j.fss.2014.02.004
[26] 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
95-110. https://doi.org/10.1016/j.ins.2020.12.053
[27] A. W. Marshall, I. Olkin, A multivariate exponential distribution, Journal of the American Statistical Association,
62 (1967), 30-44. https://doi.org/10.2307/2282907
62 (1967), 30-44. https://doi.org/10.2307/2282907
[28] R. B. Nelsen, An introduction to copulas, Springer, New York, 2006. https://link.springer.com/book/10.
1007/0-387-28678-0
1007/0-387-28678-0
[29] R. B. Nelsen, J. H. 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 (2008), 473-483. https:
//doi.org/10.1016/j.insmatheco.2006.11.011
and quasi-copulas with given diagonal sections, Insurance Mathematics and Economics, 42 (2008), 473-483. https:
//doi.org/10.1016/j.insmatheco.2006.11.011
[30] J. A. Rodr´ıguez-Lallena, M. ´Ubeda-Flores, Some new characterizations and properties of quasi-copulas, Fuzzy Sets
and Systems, 160 (2009), 717-725. https://doi.org/10.1016/j.fss.2008.02.007
and Systems, 160 (2009), 717-725. https://doi.org/10.1016/j.fss.2008.02.007
[31] A. Sancetta, S. E. Satchell, The bernstein copula and its applications to modeling and approximations of multivariate
distributions, Econometric Theory, 20 (2004), 535-562. https://doi.org/10.1017/S026646660420305X
distributions, Econometric Theory, 20 (2004), 535-562. https://doi.org/10.1017/S026646660420305X
[32] M. Sklar, Fonctions de repartition a n-dimensions et leurs marges, Publications de l’Institut de statistique de
l’Universite de Paris, 8 (1959), 229-231. https://hal.science/hal-04094463/document
l’Universite de Paris, 8 (1959), 229-231. https://hal.science/hal-04094463/document
[33] 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
sections, Fuzzy Sets and Systems, 473 (2023), 108720. https://doi.org/10.1016/j.fss.2023.108720
[34] C. Z. Yao, M. J. Li, GARCH-MIDAS-GAS-copula model for CoVaR and risk spillover in stock markets, North
American Journal of Economics and Finance, 66 (2023), 101910. https://doi.org/10.1016/j.najef.2023.
101910
American Journal of Economics and Finance, 66 (2023), 101910. https://doi.org/10.1016/j.najef.2023.
101910
[35] M. H. Zhang, Modelling total tail dependence along diagonals, Insurance: Mathematics and Economics, 42 (2008),
73-80. https://doi.org/10.1016/j.insmatheco.2007.01.002
73-80. https://doi.org/10.1016/j.insmatheco.2007.01.002
[36] 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
sections, Fuzzy Sets and Systems, 441 (2022), 335-365. https://doi.org/10.1016/j.fss.2021.12.008
)