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Characterizations of Archimedean \(n\)-copulas. (English) Zbl 1340.62054
Summary: We present three characterizations of \(n\)-dimensional Archimedean copulas: algebraic, differential and diagonal. The first is due to Jouini and Clemen. We formulate it in a more general form, in terms of an \(n\)-variable operation derived from a binary operation. The second characterization is in terms of first order partial derivatives of the copula. The last characterization uses diagonal generators, which are “regular” diagonal sections of copulas, enabling one to recover the copulas by means of an asymptotic representation.

MSC:
62H20 Measures of association (correlation, canonical correlation, etc.)
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[1] Alsina, C., Nelsen, R. B., Schweizer, B.: On the characterization of a class of binary operations on distribution functions. Statist. Probab. Lett. 17 (1993), 85-89. · Zbl 0798.60023
[2] Cuculescu, I., Theodorescu, R.: Copulas: diagonals, tracks. Rev. Roumaine Math. Pures Appl. 46 (2001), 731-742. · Zbl 1032.60009
[3] Dudek, W. A., Trokhimenko, V. S.: Menger algebras of multiplace functions. Universitatea de Stat din Moldova, Chişinău, 2006 · Zbl 1115.08001
[4] Durante, F., Sempi, C.: Copula theory: an introduction. Workshop on Copula Theory and its Applications (P. Jaworski et al., Lecture Notes in Statist. Proc. 198, Springer 2010, pp. 3-31.
[5] sciences, Encyclopedia of statistical: Vol. 2, second edition. Wiley 2006, pp. 1363-1367.
[6] Fang, K. T., Fang, B. Q.: Some families of multivariate symmetric distributions related to exponential distribution. J. Multivariate Anal. 24 (1998), 109-122. · Zbl 0635.62035
[7] Feller, W.: An introduction to probability theory and its applications. Vol. II, second edition. Wiley, New York 1971. · Zbl 0219.60003
[8] Genest, C., MacKay, J.: Copules archimédiennes et familles des lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14 (1986), 145-159. · Zbl 0605.62049
[9] Genest, C., MacKay, J.: The joy of copulas: Bivariate distributions with uniform marginals. Amer. Statist. 40 (1986), 280-285.
[10] Genest, C., Quesada-Molina, L. J., Rodríguez-Lallena, J. A., Sempi, C.: A characterization of quasicopulas. J. Multivariate Anal. 69 (1999), 193-205. · Zbl 0935.62059
[11] Gluskin, L. M.: Positional operatives. Dokl. Akad. Nauk SSSR 157 (1964), 767-770 · Zbl 0294.08001
[12] Gluskin, L. M.: Positional operatives. Mat. Sb. (N.S.) 68 (110) (1965), 444-472 · Zbl 0294.08001
[13] Gluskin, L. M.: Positional operatives. Dokl. Akad. Nauk SSSR 182 (1968), 1000-1003 · Zbl 0294.08001
[14] Hutchinson, T. P., Lai, C. D.: Continuous bivariate distributions. Emphasising applications. Rumsby Scientific, Adelaide 1990. · Zbl 1170.62330
[15] Jaworski, P.: On copulas and their diagonals. Inform. Sci. 179 (2009), 2863-2871. · Zbl 1171.62332
[16] Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997. <a href=”http://dx.doi.org/10.1002/(sici)1097-0258(19980930)17:183.0.co;2-r” target=”_blank”>DOI 10.1002/(sici)1097-0258(19980930)17:183.0.co;2-r | · Zbl 0990.62517
[17] Jouini, M. N., Clemen, R. T.: Copula models for aggregating expert opinions. Oper. Research 44 (1996), 444-457. · Zbl 0864.90067
[18] Kimberling, C. H.: A probabilistic interpretation of complete monotonicity. Aequationes Math. 10 (1974), 152-164. · Zbl 0309.60012
[19] Kuczma, M.: Functional equations in a single variable. Monografie Mat. 46, PWN, Warszawa 1968. · Zbl 0725.39003
[20] Ling, C. H.: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189-212. · Zbl 0137.26401
[21] McNeil, A. J., Nešlehová, J.: Multivariate Archimedean copulas, \(d\)-monotone functions and \(l_1\)-norm symmetric distributions. Ann. Statist. 37 (2009), 3059-3097. · Zbl 1173.62044
[22] Nelsen, R. B.: An introduction to copulas. Springer, 2006. · Zbl 1152.62030
[23] Nelsen, R. B., Quesada-Molina, J. J., Rodríguez-Lallena, J. A., Úbeda-Flores, M.: Multivariate Archimedean quasi-copulas. Distributions with given Marginals and Statistical Modelling. Kluwer, 2002, pp. 179-185. · Zbl 1135.62338
[24] Rüschendorf, L.: Mathematical risk analysis. Dependence, risk bounds, optimal allocations and portfolios. Springer, 2013 · Zbl 1266.91001
[25] Stupňanová, A., Kolesárová, A.: Associative \(n\)-dimensional copulas. Kybernetika 47 (2011), 93-99. · Zbl 1225.03071
[26] Sungur, E. A., Yang, Y.: Diagonal copulas of Archimedean class. Comm. Statist. Theory Methods 25 (1996), 1659-1676. · Zbl 0900.62339
[27] Williamson, R. E.: Multiple monotone functions and their Laplace transforms. Duke Math. J. 23 (1956), 189-207. · Zbl 0070.28501
[28] Wysocki, W.: Constructing Archimedean copulas from diagonal sections. Statist. Probab. Lett. 82 (2012), 818-826. · Zbl 1242.62041
[29] Wysocki, W.: When a copula is archimax. Statist. Probab. Lett. 83 (2013), 37-45. · Zbl 1242.62041
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