Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Archimedean operation; additive generator; diagonal generator; multiplicative generator; (Archimedean) $n$-copula; (Archimedean) $n$-quasicopula
Archimedean operation; additive generator; diagonal generator; multiplicative generator; (Archimedean) $n$-copula; (Archimedean) $n$-quasicopula
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.
References:
[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. DOI 10.1016/0167-7152(93)90001-y | MR 1223530 | Zbl 0798.60023
[2] Cuculescu, I., Theodorescu, R.: Copulas: diagonals, tracks. Rev. Roumaine Math. Pures Appl. 46 (2001), 731-742. MR 1929521 | Zbl 1032.60009
[3] Dudek, W. A., Trokhimenko, V. S.: Menger algebras of multiplace functions. Universitatea de Stat din Moldova, Chişinău, 2006 (in Russian). MR 2292134 | Zbl 1115.08001
[4] Durante, F., Sempi, C.: Copula theory: an introduction. In: Workshop on Copula Theory and its Applications (P. Jaworski et al. eds.), Lecture Notes in Statist. Proc. 198, Springer 2010, pp. 3-31. DOI 10.1007/978-3-642-12465-5_1 | MR 3051261
[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. DOI 10.1016/0047-259x(88)90105-4 | MR 0925133 | Zbl 0635.62035
[7] Feller, W.: An introduction to probability theory and its applications. Vol. II, second edition. Wiley, New York 1971. MR 0270403
[8] Genest, C., MacKay, J.: Copules archimédiennes et familles des lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14 (1986), 145-159. DOI 10.2307/3314660 | MR 0849869
[9] Genest, C., MacKay, J.: The joy of copulas: Bivariate distributions with uniform marginals. Amer. Statist. 40 (1986), 280-285. DOI 10.1080/00031305.1986.10475414 | MR 0866908
[10] Genest, C., Quesada-Molina, L. J., Rodríguez-Lallena, J. A., Sempi, C.: A characterization of quasicopulas. J. Multivariate Anal. 69 (1999), 193-205. DOI 10.1006/jmva.1998.1809 | MR 1703371
[11] Gluskin, L. M.: Positional operatives. Dokl. Akad. Nauk SSSR 157 (1964), 767-770 (in Russian). MR 0164915 | Zbl 0294.08001
[12] Gluskin, L. M.: Positional operatives. Mat. Sb. (N.S.) 68 (110) (1965), 444-472 (in Russian). MR 0193040 | Zbl 0294.08001
[13] Gluskin, L. M.: Positional operatives. Dokl. Akad. Nauk SSSR 182 (1968), 1000-1003 (in Russian). MR 0240233 | Zbl 0294.08001
[14] Hutchinson, T. P., Lai, C. D.: Continuous bivariate distributions. Emphasising applications. Rumsby Scientific, Adelaide 1990. MR 1070715 | Zbl 1170.62330
[15] Jaworski, P.: On copulas and their diagonals. Inform. Sci. 179 (2009), 2863-2871. DOI 10.1016/j.ins.2008.09.006 | MR 2547755 | Zbl 1171.62332
[16] Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London 1997. DOI 10.1002/(sici)1097-0258(19980930)17:18<2154::aid-sim913>3.0.co;2-r | MR 1462613 | Zbl 0990.62517
[17] Jouini, M. N., Clemen, R. T.: Copula models for aggregating expert opinions. Oper. Research 44 (1996), 444-457. DOI 10.1287/opre.44.3.444 | Zbl 0864.90067
[18] Kimberling, C. H.: A probabilistic interpretation of complete monotonicity. Aequationes Math. 10 (1974), 152-164. DOI 10.1007/bf01832852 | MR 0353416 | Zbl 0309.60012
[19] Kuczma, M.: Functional equations in a single variable. Monografie Mat. 46, PWN, Warszawa 1968. MR 0228862 | Zbl 0725.39003
[20] Ling, C. H.: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189-212. MR 0190575 | 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. DOI 10.1214/07-aos556 | MR 2541455
[22] Nelsen, R. B.: An introduction to copulas. Springer, 2006. DOI 10.1007/0-387-28678-0 | MR 2197664 | Zbl 1152.62030
[23] Nelsen, R. B., Quesada-Molina, J. J., Rodr{í}guez-Lallena, J. A., Úbeda-Flores, M.: Multivariate Archimedean quasi-copulas. In: Distributions with given Marginals and Statistical Modelling. Kluwer, 2002, pp. 179-185. DOI 10.1007/978-94-017-0061-0_19 | MR 2058991 | Zbl 1135.62338
[24] Rüschendorf, L.: Mathematical risk analysis. Dependence, risk bounds, optimal allocations and portfolios. Springer, 2013 (Chapter 1). DOI 10.1007/978-3-642-33590-7 | MR 3051756 | Zbl 1266.91001
[25] Stupňanová, A., Kolesárová, A.: Associative $n$-dimensional copulas. Kybernetika 47 (2011), 93-99. MR 2807866 | Zbl 1225.03071
[26] Sungur, E. A., Yang, Y.: Diagonal copulas of Archimedean class. Comm. Statist. Theory Methods 25 (1996), 1659-1676. DOI 10.1080/03610929608831791 | MR 1411104 | Zbl 0900.62339
[27] Williamson, R. E.: Multiple monotone functions and their Laplace transforms. Duke Math. J. 23 (1956), 189-207. DOI 10.1215/s0012-7094-56-02317-1 | MR 0077581
[28] Wysocki, W.: Constructing Archimedean copulas from diagonal sections. Statist. Probab. Lett. 82 (2012), 818-826. DOI 10.1016/j.spl.2012.01.008 | MR 2899525 | Zbl 1242.62041
[29] Wysocki, W.: When a copula is archimax. Statist. Probab. Lett. 83 (2013), 37-45. DOI 10.1016/j.spl.2012.01.008 | MR 2998721
Search
Partner of
