New results on perturbation-based copulas

References

[1] Abel, N. H. (1826). Untersuchung der Functionen zweier unabhängig veränderlichen Größen x und y, wie f (x, y), welche die Eigenschaft haben, daß f (z, f (x, y)) eine symmetrische Function von z, x und y ist. J. Reine Angew. Math. 1, 11–15.Search in Google Scholar

[2] Albanese A. and C. Sempi (2016). Idempotent copulæ: Ordinal sums and Archimedean copulæ. J. Math. Anal. Appl. 438(2), 1055–1065.10.1016/j.jmaa.2016.02.037Search in Google Scholar

[3] Alsina, C., M. J. Frank, and B. Schweizer (2006). Associative Functions. World Scientific Publishing, Singapore.Search in Google Scholar

[4] Amblard, C. and S. Girard (2009). A new extension of bivariate FGM copulas. Metrika 70, 1–17.10.1007/s00184-008-0174-7Search in Google Scholar

[5] Anakkamatee, W., S. Dhompongsa, and S. Tasena (2014). A constructive proof of the Sklar’s theorem on copulas. J. Nonlinear Convex Anal. 15(6), 1137–1145.Search in Google Scholar

[6] Arias-García, J. J., H. De Meyer, and B. De Baets (2018). On the construction of radially symmetric copulas in higher dimensions. Fuzzy Set. Syst. 335, 30–47.10.1016/j.fss.2017.11.004Search in Google Scholar

[7] Arnold, V. I (1989). Mathematical Methods of Classical Mechanics. Second edition. Springer, New York.10.1007/978-1-4757-2063-1Search in Google Scholar

[8] Bahraoui, T. and J.-F. Quessy (2017). Tests of radial symmetry for multivariate copulas based on the copula characteristic function. Electron. J. Stat. 11(1), 2066–2096.10.1214/17-EJS1280Search in Google Scholar

[9] Bellman, R. (1964). Perturbation Techniques in Mathematics, Physics, and Engineering. Holt, Rinehart and Winston, New York.Search in Google Scholar

[10] Birkhoff, G. (1973). Lattice Theory. Third edition. American Mathematical Society, Providence RI.Search in Google Scholar

[11] Blomqvist, N. (1950). On a measure of dependence between two random variables. Ann. Math. Statist. 21(4), 593–600.10.1214/aoms/1177729754Search in Google Scholar

[12] Bransden, B. H. and C. J. Joachain (2000). Quantum Mechanics. Second edition. Pearson Education, Harlow.Search in Google Scholar

[13] Cambanis, S. (1977). Some properties and generalizations of multivariate Eyraud-Gumbel-Morgenstern distributions. J. Multivariate Anal. 7(4), 551–559.10.1016/0047-259X(77)90066-5Search in Google Scholar

[14] Cambanis, S. (1991). On Eyraud-Farlie-Gumbel-Morgenstern random processes. In G. Dall’Aglio, S. Kotz, and G. Salinetti (Eds.), Advances in Probability Distributions with Given Marginals, pp. 207–222. Kluwer Academic Publishers, Dordrecht.10.1007/978-94-011-3466-8_11Search in Google Scholar

[15] Clifford, A. H. (1954). Naturally totally ordered commutative semigroups. Amer. J. Math. 76(3), 631–646.10.2307/2372706Search in Google Scholar

[16] Clifford, A. H. (1958). Connected ordered topological semigroups with idempotent endpoints. I. Trans. Amer. Math. Soc. 88(1), 80–98.Search in Google Scholar

[17] Clifford, A. H. (1958). Totally ordered commutative semigroups. Bull. Amer. Math. Soc. 64(6), 305–316.10.1090/S0002-9904-1958-10221-9Search in Google Scholar

[18] Clifford, A. H. (1959). Connected ordered topological semigroups with idempotent endpoints. II. Trans. Amer. Math. Soc. 91(2), 193–208.Search in Google Scholar

[19] Cooray, K. (2019). A new extension of the FGM copula for negative association. Comm. Statist. Theory Methods 48(8), 1902–1919.10.1080/03610926.2018.1440312Search in Google Scholar

[20] Cuadras, C. M. (2009). Constructing copula functions with weighted geometric means. J. Statist. Plann. Inference 139(11), 3766–3772.10.1016/j.jspi.2009.05.016Search in Google Scholar

[21] Cuadras, C. M. (2015). Contributions to the diagonal expansion of a bivariate copula with continuous extensions. J. Multivariate Anal. 139, 28–44.10.1016/j.jmva.2015.02.015Search in Google Scholar

[22] Cuadras, C. M. and J. Augé (1981). A continuous general multivariate distribution and its properties. Comm. Statist. Theory Methods 10(4), 339–353.10.1080/03610928108828042Search in Google Scholar

[23] Dall’Aglio, G. (1956). Sugli estremi dei momenti delle funzioni di ripartizione doppia. Ann. Sc. Normale Super. Pisa Cl. Sci. (3)10, 35–74.Search in Google Scholar

[24] Dall’Aglio, G. (1959). Sulla compatibilità delle funzioni di ripartizione doppia. Rend. Mat. Appl. (3-4)18, 385–413.Search in Google Scholar

[25] Dall’Aglio, G. (1960). Les fonctions extrêmes de la classe de Fréchet à 3 dimensions. Publ. Inst. Statist. Univ. Paris 9, 175–188.Search in Google Scholar

[26] Darsow, W. F., B. Nguyen, and E. T. Olsen (1992). Copulas and Markov processes. Illinois J. Math. 36(4), 600–642.10.1215/ijm/1255987328Search in Google Scholar

[27] Darsow, W. F. and E. T. Olsen (2010). Characterization of idempotent 2-copulas. Note Mat. 30(1), 147–177.Search in Google Scholar

[28] De Baets, B. and H. De Meyer (2007). Orthogonal grid constructions of copulas. IEEE Trans. Fuzzy Syst. 15(6), 1053–1062.10.1109/TFUZZ.2006.890681Search in Google Scholar

[29] De Baets, B., H. De Meyer, and T. Jwaid (2019). On the degree of asymmetry of a quasi-copula with respect to a curve. Fuzzy Set. Syst. 354, 84–103.10.1016/j.fss.2018.05.002Search in Google Scholar

[30] De Baets, B., H. De Meyer, J. Kalická, and R. Mesiar (2009). Flipping and cyclic shifting of binary aggregation functions. Fuzzy Set. Syst. 160(6), 752–765.10.1016/j.fss.2008.03.008Search in Google Scholar

[31] De Baets, B., H. De Meyer, and M. Úbeda-Flores (2009). Opposite diagonal sections of quasi-copulas and copulas. Internat. J. Uncertain. Fuzz. 17(4), 481–490.10.1142/S0218488509006108Search in Google Scholar

[32] Dirac, P. A. M. (1927). The quantum theory of the emission and absorption of radiation. Proc. Roy. Soc. London Ser. A 114(767), 243–265.10.1098/rspa.1927.0039Search in Google Scholar

[33] Durante, F., J. Fernández-Sánchez, and C. Sempi (2012). Sklar’s theorem obtained via regularization techniques. Nonlinear Anal. 75(2), 769–774.10.1016/j.na.2011.09.006Search in Google Scholar

[34] Durante, F., J. Fernández-Sánchez, and C. Sempi (2013). A topological proof of Sklar’s theorem. Appl. Math. Lett. 26(9), 945–948.10.1016/j.aml.2013.04.005Search in Google Scholar

[35] Durante, F., J. Fernández-Sánchez, and C. Sempi (2013). How to prove Sklar’s theorem. In H. Bustince, J. Fernandez, R. Mesiar, and T. Calvo (Eds.), Aggregation Functions in Theory and in Practise, pp. 85–90. Springer, Heidelberg.10.1007/978-3-642-39165-1_12Search in Google Scholar

[36] Durante, F., J. Fernández-Sánchez, and M. Úbeda-Flores (2013). Bivariate copulas generated by perturbations. Fuzzy Set. Syst. 228, 137–144.10.1016/j.fss.2012.08.008Search in Google Scholar

[37] Durante, F., S. Saminger-Platz, and P. Sarkoci (2008). On representations of 2-increasing binary aggregation functions. Inform. Sci. 178(23), 4534–4541.10.1016/j.ins.2008.08.004Search in Google Scholar

[38] Durante, F., S. Saminger-Platz, and P. Sarkoci (2009). Rectangular patchwork for bivariate copulas and tail dependence. Comm. Statist. Theory Methods 38(15), 2515–2527.10.1080/03610920802571203Search in Google Scholar

[39] Durante, F. and C. Sempi (2016). Principles of Copula Theory. CRC Press, Boca Raton FL.Search in Google Scholar

[40] Dvořák, A., M. Holčapek, and J. Paseka (2021). On ordinal sums of partially ordered monoids: A unified approach to ordinal sum constructions of t-norms, t-conorms and uninorms. Fuzzy Set. Syst., to appear. Available at https://doi.org/10.1016/j.fss.2021.04.008.10.1016/j.fss.2021.04.008Search in Google Scholar

[41] Eyraud, H. (1936). Les principes de la mesure des corrélations. Ann. Univ. Lyon, Sect. A 1, 30–47.Search in Google Scholar

[42] Farlie, D. J. G. (1960). The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47(3-4), 307–323.10.1093/biomet/47.3-4.307Search in Google Scholar

[43] Faugeras, O. P. (2013). Sklar’s theorem derived using probabilistic continuation and two consistency results. J. Multivariate Anal. 122, 271–277.10.1016/j.jmva.2013.07.010Search in Google Scholar

[44] Fernández-Sánchez, J., W. Trutschnig, and M. Tschimpke (2021). Markov product invariance in classes of bivariate copulas characterized by univariate functions. J. Math. Anal. Appl. 501(2), Article ID 125184, 15 pages.10.1016/j.jmaa.2021.125184Search in Google Scholar

[45] Fernández-Sánchez, J. and M. Úbeda-Flores (2018). Constructions of copulas with given diagonal (and opposite diagonal) sections and some generalizations. Depend. Model. 6, 139–155.10.1515/demo-2018-0009Search in Google Scholar

[46] Fernández-Sánchez, J. and M. Úbeda-Flores (2018). Proving Sklar’s theorem via Zorn’s lemma. Internat. J. Uncertain. Fuzz. 26(1), 81–85.10.1142/S0218488518500058Search in Google Scholar

[47] Fernández-Sánchez, J. and M. Úbeda-Flores (2019). Solution to two open problems on perturbations of the product copula. Fuzzy Set. Syst. 354, 116–122.10.1016/j.fss.2018.06.013Search in Google Scholar

[48] Féron, R. (1956). Sur les tableaux de corrélation dont les marges sont données. Cas de l’espace à trois dimensions. Publ. Inst. Statist. Univ. Paris 5, 3–12.Search in Google Scholar

[49] Frank, M. J. (1979). On the simultaneous associativity of F(x, y) and x + y − F(x, y). Aequationes Math. 19, 194–226.10.1007/BF02189866Search in Google Scholar

[50] Fréchet, M. (1951). Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon, Sect. A(3) 14, 53–77.Search in Google Scholar

[51] Fréchet, M. (1958). Remarques au sujet de la note précédente. C. R. Acad. Sci. Paris 246, 2719–2720.Search in Google Scholar

[52] Fredricks, G. A. and R. B. Nelsen (1997). Copulas constructed from diagonal sections. In V. Beneš and J. Štěpán (Eds.), Distributions with Given Marginals and Moment Problems, pp. 129–136. Springer, Dordrecht.10.1007/978-94-011-5532-8_16Search in Google Scholar

[53] Fredricks, G. A. and R. B. Nelsen (2002). The Bertino family of copulas. In C. M. Cuadras, J. Fortiana, and J. A. Rodríguez-Lallena (Eds.), Distributions with Given Marginals and Statistical Modelling, pp. 81–91. Springer, Dordrecht.10.1007/978-94-017-0061-0_10Search in Google Scholar

[54] Fredricks, G. A. and R. B. Nelsen (2007). On the relationship between Spearman’s rho and Kendall’s tau for pairs of continuous random variables. J. Statist. Plann. Inference 137(7), 2143–2150.10.1016/j.jspi.2006.06.045Search in Google Scholar

[55] Genest, C. and J. G. Nešlehová (2014). On tests of radial symmetry for bivariate copulas. Statist. Papers 55, 1107–1119.10.1007/s00362-013-0556-4Search in Google Scholar

[56] Gini, C. (1955). Variabilità e mutabilità. In E. Pizetti and T. Salvemini (Eds.), Memorie di Metodologica Statistica. Libreria Eredi Virgilio Veschi, Roma.Search in Google Scholar

[57] Gudendorf, G. and J. Segers (2010). Extreme-value copulas. In P. Jaworski, F. Durante, W. K. Härdle, and T. Rychlik (Eds.), Copula Theory and Its Applications, pp. 127–145. Springer, Berlin.10.1007/978-3-642-12465-5_6Search in Google Scholar

[58] Gumbel, E. J. (1958). Distributions à plusieurs variables dont les marges sont données. C. R. Acad. Sci. Paris 246, 2717–2719.Search in Google Scholar

[59] Gutzwiller, M. C. (1998). Moon-Earth-Sun: The oldest three-body problem. Rev. Modern Phys. 70, 589–639.10.1103/RevModPhys.70.589Search in Google Scholar

[60] Hoeffding, W. (1940) Maßstabinvariante Korrelationstheorie. Schr. Math. Inst. Inst. Angew. Math. Univ. Berlin 5, 181–233. Also in Hoeffding, W. (1994). Scale-invariant correlation theory. In N. I. Fisher and P. K. Sen (Eds.), The Collected Works of Wassily Hoeffding, pp. 57–107. Springer, New York.10.1007/978-1-4612-0865-5_4Search in Google Scholar

[61] Hoeffding, W. (1941). Maßstabinvariante Korrelationsmaße für diskontinuierliche Verteilungen. Arch. Math. Wirtsch.-Sozialforschg. 7, 49–70. Also in Hoeffding, W. (1994). Scale-invariant correlation theory. In N. I. Fisher and P. K. Sen (Eds.), The Collected Works of Wassily Hoeffding, pp. 109–133. Springer, New York.10.1007/978-1-4612-0865-5_5Search in Google Scholar

[62] Hürlimann, W. (2017). A comprehensive extension of the FGM copula. Statist. Papers 58, 373–392.10.1007/s00362-015-0703-1Search in Google Scholar

[63] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.Search in Google Scholar

[64] Joe, H. (2015). Dependence Modeling with Copulas. CRC Press, Boca Raton FL.Search in Google Scholar

[65] Kendall, M. G. (1938). A new measure of rank correlation. Biometrika 30(1-2), 81–93.10.1093/biomet/30.1-2.81Search in Google Scholar

[66] Kepler, J. (1609). Astronomia Nova. Prague.Search in Google Scholar

[67] Kepler, J. (1618-1621). Epitome Astronomiae Copernicanae. Linz.Search in Google Scholar

[68] Kim, J.-M., E. A. Sungur, T. Choi, and T.-Y. Heo (2011). Generalized bivariate copulas and their properties. Model Assist. Stat. Appl. 6(2), 127–136.10.3233/MAS-2011-0185Search in Google Scholar

[69] Klement, E. P., A. Kolesárová, R. Mesiar, and S. Saminger-Platz (2017). On the role of ultramodularity and Schur concavity in the construction of binary copulas. J. Math. Inequal. 11(2), 361–381.10.7153/jmi-11-32Search in Google Scholar

[70] Klement, E. P., A. Kolesárová, R. Mesiar, and C. Sempi (2007). Copulas constructed from horizontal sections. Comm. Statist. Theory Methods 36(11), 2901–2911.10.1080/03610920701386976Search in Google Scholar

[71] Klement, E. P. and R. Mesiar (2006). How non-symmetric can a copula be? Comment. Math. Univ. Carolin. 47(1), 141–148.Search in Google Scholar

[72] Klement, E. P., R. Mesiar, and E. Pap (2000). Triangular Norms. Springer, Dordrecht.10.1007/978-94-015-9540-7Search in Google Scholar

[73] Klement, E. P., R. Mesiar, and E. Pap (2001). Uniform approximation of associative copulas by strict and non-strict copulas. Illinois J. Math. 45(4), 1393–1400.10.1215/ijm/1258138075Search in Google Scholar

[74] Klement, E. P., R. Mesiar, and E. Pap (2002). Triangular norms as ordinal sums of semigroups in the sense of A. H. Clifford. Semigroup Forum 65, 71–82.10.1007/s002330010127Search in Google Scholar

[75] Kolesárová, A., G. Mayor, and R. Mesiar (2015). Quadratic constructions of copulas. Inform. Sci. 310, 69–76.10.1016/j.ins.2015.03.016Search in Google Scholar

[76] Kolesárová, A., R. Mesiar, and J. Kalická (2013). On a new construction of 1-Lipschitz aggregation functions, quasi-copulas and copulas. Fuzzy Set. Syst. 226, 19–31.10.1016/j.fss.2013.01.005Search in Google Scholar

[77] Komorník, J., M. Komorníková, and J. Kalická (2017). Dependence measures for perturbations of copulas. Fuzzy Set. Syst. 324, 100–116.10.1016/j.fss.2017.01.014Search in Google Scholar

[78] Komorník, J., M. Komorníková, and J. Kalická (2018). Families of perturbation copulas generalizing the FGM family and their relations to dependence measures. In V. Torra, R. Mesiar, and B. De Baets (Eds.), Aggregation Functions in Theory and in Practice, pp. 53–63. Springer, Cham.10.1007/978-3-319-59306-7_6Search in Google Scholar

[79] Komorník, J., M. Komorníková, J. Kalická, and C. Nguyen (2016). Tail dependence of perturbed copulas. J. Stat. Theory Appl. 15(2), 153–160.10.2991/jsta.2016.15.2.5Search in Google Scholar

[80] Lagrange, J.-L. (1788). Méchanique Analitique. La Veuve Desaint, Paris.Search in Google Scholar

[81] Laplace, P.-S. (1798–1825). Traité de Mécanique Céleste. L’Imprimerie de Crapelet, Paris.Search in Google Scholar

[82] Lee, M.-L. T. (1996). Properties and applications of the Sarmanov family of bivariate distributions. Comm. Statist. Theory Methods 25(6), 1207–1222.10.1080/03610929608831759Search in Google Scholar

[83] Ling, C.-H. (1965). Representation of associative functions. Publ. Math. Debrecen 12, 189–212.10.5486/PMD.1965.12.1-4.19Search in Google Scholar

[84] Mardia, K. V. (1970). Families of Bivariate Distributions. Charles Griffin, London.Search in Google Scholar

[85] Marinacci, M. and L. Montrucchio (2005). Ultramodular functions. Math. Oper. Res. 30(2), 311–332.10.1287/moor.1040.0143Search in Google Scholar

[86] Mesiar, R., J. Komorník, and M. Komorníková (2013). On some construction methods for bivariate copulas. In H. Bustince, J. Fernandez, R. Mesiar, and T. Calvo (Eds.), Aggregation Functions in Theory and in Practise, pp. 39–45. Springer, Heidelberg.10.1007/978-3-642-39165-1_7Search in Google Scholar

[87] Mesiar, R., M. Komorníková, and J. Komorník (2015). Perturbation of bivariate copulas. Fuzzy Set. Syst. 268, 127–140.10.1016/j.fss.2014.04.016Search in Google Scholar

[88] Mesiar, R. and V. Najjari (2014). New families of symmetric/asymmetric copulas. Fuzzy Set. Syst. 252, 99–110.10.1016/j.fss.2013.12.015Search in Google Scholar

[89] Mesiar, R., A. Sheikhi, and M. Komorníková (2019). Random noise and perturbation of copulas. Kybernetika 55(2), 422–434.10.14736/kyb-2019-2-0422Search in Google Scholar

[90] Mesiar, R. and J. Szolgay (2004). W-ordinal sums of copulas and quasi-copulas. Proceedings MAGIA & UWPM 2004, pp. 78–83. Publishing House of Slovak University of Technology, Bratislava.Search in Google Scholar

[91] Mikami, T. (1997). Large deviations and central limit theorems for Eyraud-Farlie-Gumbel-Morgenstern processes. Statist. Probab. Lett. 35(1), 73–78.10.1016/S0167-7152(96)00218-0Search in Google Scholar

[92] Morgenstern, D. (1956). Einfache Beispiele zweidimensionaler Verteilungen. Mitteilungsbl. Math. Statist. 8, 234–235.Search in Google Scholar

[93] Nelsen, R. B. (2006). An Introduction to Copulas. Second edition. Springer, New York.Search in Google Scholar

[94] Nelsen, R. B. (2007). Extremes of nonexchangeability. Statist. Papers 48, 329–336.10.1007/s00362-006-0336-5Search in Google Scholar

[95] Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. Jussu Societatis Regiae ac typis Josephi Streater, London.10.5479/sil.52126.39088015628399Search in Google Scholar

[96] Oertel, F. (2015). An analysis of the Rüschendorf transform - with a view towards Sklar’s Theorem. Depend. Model. 3, 113–125.10.1515/demo-2015-0008Search in Google Scholar

[97] Olsen, E. T., W. F. Darsow, and B. Nguyen (1996). Copulas and Markov operators. In L. Rüschendorf, B. Schweizer, and M. D. Taylor (Eds.), Distributions with Fixed Marginals and Related Topics, pp. 244–259. Institute of Mathematical Statistics, Hayward CA.10.1214/lnms/1215452623Search in Google Scholar

[98] Rodríguez-Lallena, J. A. and M. Úbeda-Flores (2004). A new class of bivariate copulas. Statist. Probab. Lett. 66(3), 315–325.10.1016/j.spl.2003.09.010Search in Google Scholar

[99] Rodríguez-Lallena, J. A. and M. Úbeda-Flores (2005). Best-possible bounds on sets of multivariate distribution functions. Comm. Statist. Theory Methods 33(4), 805–820.10.1081/STA-120028727Search in Google Scholar

[100] Rüschendorf, L. (2009). On the distributional transform, Sklar’s theorem, and the empirical copula process. J. Statist. Plann. Inference 139(11), 3921–3927.10.1016/j.jspi.2009.05.030Search in Google Scholar

[101] Saminger, S. (2006). On ordinal sums of triangular norms on bounded lattices. Fuzzy Set. Syst. 157(10), 1403–1416.10.1016/j.fss.2005.12.021Search in Google Scholar

[102] Saminger-Platz, S., M. Dibala, E. P. Klement, and R. Mesiar (2017). Ordinal sums of binary conjunctive operations based on the product. Publ. Math. Debrecen 91(1-2), 63–80.10.5486/PMD.2017.7636Search in Google Scholar

[103] Saminger-Platz, S., A. Kolesárová, R. Mesiar, and E. P. Klement (2020). The key role of convexity in some copula constructions. Eur. J. Math. 6, 533–560.10.1007/s40879-019-00346-3Search in Google Scholar

[104] Saminger-Platz, S., A. Kolesárová, A. Šeliga, R. Mesiar, and E. P. Klement (2021). The impact on the properties of the EFGM copulas when extending this family. Fuzzy Set. Syst. 415, 1–26.10.1016/j.fss.2020.11.001Search in Google Scholar

[105] Sarmanov, O. V. (1966). Generalized normal correlation and two-dimensional Fréchet classes. Dokl. Akad. Nauk SSSR 168(1), 32–35.Search in Google Scholar

[106] Schreyer, M., R. Paulin, and W. Trutschnig (2017). On the exact region determined by Kendall’s τ and Spearman’s ϱ. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79(2), 613–633.10.1111/rssb.12181Search in Google Scholar

[107] Schur, I. (1923). Über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. S.-B. Berlin. Math. Ges. 22, 9–20.Search in Google Scholar

[108] Schwarz, G. (1985). Multivariate distributions with uniformly distributed projections. Ann. Probab. 13(4), 1371–1372.10.1214/aop/1176992821Search in Google Scholar

[109] Schweizer, B. and A. Sklar (1958). Espaces métriques aléatoires. C. R. Acad. Sci. Paris 247, 2092–2094.Search in Google Scholar

[110] Schweizer, B. and A. Sklar (1960). Statistical metric spaces. Pacific J. Math. 10(1), 313–334.10.2140/pjm.1960.10.313Search in Google Scholar

[111] Schweizer, B. and A. Sklar (1961). Associative functions and statistical triangle inequalities. Publ. Math. Debrecen 8, 169–186.Search in Google Scholar

[112] Schweizer, B. and A. Sklar (1963). Associative functions and abstract semigroups. Publ. Math. Debrecen 10, 69–81.Search in Google Scholar

[113] Schweizer, B. and A. Sklar (1983). Probabilistic Metric Spaces. North-Holland, New York.Search in Google Scholar

[114] Šeliga, A., M. Kauers, S. Saminger-Platz, R. Mesiar, A. Kolesárová, and E. P. Klement (2021). Polynomial bivariate copulas of degree five: characterization and some particular inequalities. Depend. Model. 9, 13–42.10.1515/demo-2021-0101Search in Google Scholar

[115] Sheikhi, A., V. Amirzadeh, and R. Mesiar (2021). A comprehensive family of copulas to model bivariate random noise and perturbation. Fuzzy Set. Syst. 415, 27–36.10.1016/j.fss.2020.04.010Search in Google Scholar

[116] Sheikhi, A., F. Arad, R. Mesiar, and L. Vavríková (2020). Random noise and perturbation of copula with a copula induced noise. Int. J. Gen. Syst. 49(8), 856–871.10.1080/03081079.2020.1786378Search in Google Scholar

[117] Shubina, M. and M.-L. T. Lee (2004). On maximum attainable correlation and other measures of dependence for the Sarmanov family of bivariate distributions. Comm. Statist. Theory Methods 33(5), 1031–1052.10.1081/STA-120029824Search in Google Scholar

[118] Siburg, K. F. and P. A. Stoimenov (2008). Gluing copulas. Comm. Statist. Theory Methods 37(19), 3124–3134.10.1080/03610920802074844Search in Google Scholar

[119] Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.Search in Google Scholar

[120] Sklar, A. (1973). Random variables, joint distribution functions, and copulas. Kybernetika 9(6), 449–460.Search in Google Scholar

[121] Spearman, C. (1904). The proof and measurement of association between two things. Am. J. Psychol. 15, 72–101.10.2307/1412159Search in Google Scholar

[122] Sriboonchitta, S. and V. Kreinovich (2018). Why are FGM copulas successful? A simple explanation. Adv. Fuzzy Syst. 2018, Article ID 5872195, 5 pages.10.1155/2018/5872195Search in Google Scholar

[123] Tchen, A. H. (1980). Inequalities for distributions with given marginals. Ann. Probab. 8(4), 814–827.10.1214/aop/1176994668Search in Google Scholar