Copula Theory: An Introduction

Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 198)

Abstract

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas.

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References

  1. 1.
    Alsina, C., Frank, M.J., Schweizer, B.: Associative functions. Triangular norms and copulas. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2006)MATHCrossRefGoogle Scholar
  2. 2.
    Alsina, C., Nelsen, R.B., Schweizer, B.: On the characterization of a class of binary operations on distribution functions. Statist. Probab. Lett. 17 (2), 85–89 (1993)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Alvoni, E., Papini, P.L., Spizzichino, F.: On a class of transformations of copulas and quasi–copulas. Fuzzy Sets and Systems 160 (3), 334–343 (2009)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Amblard, C., Girard, S.: Une famille semi-paramétrique de copules symétriques bivariées. C. R. Acad. Sci. Paris Sér. I Math. 333 (2), 129–132 (2001)MATHMathSciNetGoogle Scholar
  5. 5.
    Amblard, C., Girard, S.: Symmetry and dependence properties within a semiparametric family of bivariate copulas. J. Nonparametr. Stat. 14 (6), 715–727 (2002)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Amblard, C., Girard, S.: A new symmetric extension of FGM copulas. Metrika 70 (1), 1–17 (2009)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Barndorff-Nielsen, O.E., Lindner, A.M.: Lévy copulas: dynamics and transforms of Upsilon type. Scand. J. Statist. 34 (2), 298–316 (2007)MATHMathSciNetGoogle Scholar
  8. 8.
    Bassan, B., Spizzichino, F.: Bivariate survival models with Clayton aging functions. Insurance Math. Econom. 37 (1), 6–12 (2005)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bassan, B., Spizzichino, F.: Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes. J. Multivariate Anal. 93 (2), 313–339 (2005)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Bäuerle, N., Blatter, A., Müller, A.: Dependence properties and comparison results for Lévy processes. Math. Methods Oper. Res. 67 (1), 161–186 (2008)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Beneš, V., Štˇepán, J. (eds.): Distributions with given marginals and moment problems. Kluwer Academic Publishers, Dordrecht (1997)MATHGoogle Scholar
  12. 12.
    Berg, G.: Copula goodness-of-fit testing: an overview and power comparison. Europ. J. Finance 15 (7–8), 675–701 (2009)CrossRefGoogle Scholar
  13. 13.
    Billingsley, P.: Probability and measure, third edn. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York (1995). A Wiley-Interscience PublicationGoogle Scholar
  14. 14.
    Burchard, A., Hajaiej, H.: Rearrangement inequalities for functionals with monotone integrands. J. Funct. Anal. 233 (2), 561–582 (2006)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Carley, H.: Maximum and minimum extensions of finite subcopulas. Comm. Statist. Theory Methods 31 (12), 2151–2166 (2002)MATHMathSciNetGoogle Scholar
  16. 16.
    Carley, H., Taylor, M.D.: A new proof of Sklar’s theorem. In: C.M. Cuadras, J. Fortiana, J.A. Rodriguez-Lallena (eds.) Distributions with given marginals and statistical modelling, pp. 29–34. Kluwer Acad. Publ., Dordrecht (2002)Google Scholar
  17. 17.
    Chakak, A., Koehler, K.J.: A strategy for constructing multivariate distributions. Comm. Statist. Simulation Comput. 24 (3), 537–550 (1995)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Charpentier, A.: Dynamic dependence ordering for Archimedean copulas and distorted copulas. Kybernetika (Prague) 44 (6), 777–794 (2008)MATHMathSciNetGoogle Scholar
  19. 19.
    Cherubini, U., Luciano, E., Vecchiato, W.: Copula methods in finance. Wiley Finance Series. John Wiley & Sons Ltd., Chichester (2004)Google Scholar
  20. 20.
    Choro´s, B., Ibragimov, R., Permiakova, E.: Copula estimation. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Springer (2010).Google Scholar
  21. 21.
    Clayton, D.G.: A model for association in bivariate life tables and its application in epidemiological studies of familial dependency in chronic disease incidence. Biometrika 65, 141–151 (1978)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Cook, R.D., Johnson, M.E.: A family of distributions for modelling nonelliptically symmetric multivariate data. J. Roy. Statist. Soc. Ser. B 43 (2), 210–218 (1981)MATHMathSciNetGoogle Scholar
  23. 23.
    Crane, G., van der Hoek, J.: Using distortions of copulas to price synthetic CDOs. Insurance Math. Econom. 42 (3), 903–908 (2008)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Cuadras, C.M.: Constructing copula functions with weighted geometric means. J. Statist. Plan. Infer. 139 (11), 3766–3772 (2009) 1 Copula Theory: an Introduction 25MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Cuadras, C.M., Augé, J.: A continuous general multivariate distribution and its properties. Comm. Statist. A—Theory Methods 10 (4), 339–353 (1981)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Cuadras, C.M., Fortiana, J., Rodriguez-Lallena, J.A. (eds.): Distributions with given marginals and statistical modelling. Kluwer Academic Publishers, Dordrecht (2002). Papers from the meeting held in Barcelona, July 17–20, 2000Google Scholar
  27. 27.
    Cuculescu, I., Theodorescu, R.: Copulas: diagonals, tracks. Rev. Roumaine Math. Pures Appl. 46 (6), 731–742 (2002) (2001)MathSciNetGoogle Scholar
  28. 28.
    Czado, C.: Pair-copula constructions. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Springer (2010).Google Scholar
  29. 29.
    Dall’Aglio, G.: Sulla compatibilità delle funzioni di ripartizione doppia. Rend. Mat. e Appl. (5) 18, 385–413 (1959)MathSciNetGoogle Scholar
  30. 30.
    Dall’Aglio, G.: Fréchet classes and compatibility of distribution functions. In: Symposia Mathematica, Vol IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971), pp. 131–150. Academic Press, London (1972)Google Scholar
  31. 31.
    Dall’Aglio, G.: Fréchet classes: the beginnings. In: G. Dall’Aglio, S. Kotz, G. Salinetti (eds.) Advances in probability distributions with given marginals (Rome, 1990), Math. Appl., vol. 67, pp. 1–12. Kluwer Acad. Publ., Dordrecht (1991)Google Scholar
  32. 32.
    Dall’Aglio, G., Kotz, S., Salinetti, G. (eds.): Advances in probability distributions with given marginals, Mathematics and its Applications, vol. 67. Kluwer Academic Publishers Group, Dordrecht (1991).Google Scholar
  33. 33.
    Darsow, W.F., Nguyen, B., Olsen, E.T.: Copulas and Markov processes. Illinois J. Math. 36 (4), 600–642 (1992)MATHMathSciNetGoogle Scholar
  34. 34.
    De Baets, B., De Meyer, H.: Orthogonal grid constructions of copulas. IEEE Trans. Fuzzy Systems 15 (6), 1053–1062 (2007)CrossRefGoogle Scholar
  35. 35.
    De Baets, B., De Meyer, H., Mesiar, R.: Asymmetric semilinear copulas. Kybernetika (Prague) 43 (2), 221–233 (2007)MATHMathSciNetGoogle Scholar
  36. 36.
    De Baets, B., De Meyer, H., Úbeda-Flores, M.: Opposite diagonal sections of quasi-copulas and copulas. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 17 (4), 481–490 (2009)MATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Deheuvels, P.: Caractérisation complète des lois extrêmes multivariées et de la convergence des types extrêmes. Publ. Inst. Stat. Univ. Paris 23 (3-4), 1–36 (1978)MATHGoogle Scholar
  38. 38.
    Deheuvels, P.: Indépendance multivariée partielle et inégalités de Fréchet. In: Studies in probability and related topics, pp. 145–155. Nagard, Rome (1983)Google Scholar
  39. 39.
    Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D.: The concept of comonotonicity in actuarial science and finance: applications. Insurance Math. Econom. 31 (2), 133–161 (2002)MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R., Vyncke, D.: The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom. 31 (1), 3–33 (2002). 5th IME Conference (University Park, PA, 2001)MATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Drouet-Mari, D., Kotz, S.: Correlation and dependence. Imperial College Press, London (2001)CrossRefGoogle Scholar
  42. 42.
    Durante, F.: Construction of non-exchangeable bivariate distribution functions. Statist. Papers 50 (2), 383–391 (2009)MATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Durante, F., Foschi, R., Sarkoci, P.: Distorted copulas: constructions and tail dependence. Comm. Statist. Theory Methods (2010). In pressGoogle Scholar
  44. 44.
    Durante, F., Hofert, M., Scherer, M.: Multivariate hierarchical copulas with shocks. Methodol. Comput. Appl. Probab. (2009). In pressGoogle Scholar
  45. 45.
    Durante, F., Jaworski, P.: Absolutely continuous copulas with given diagonal sections. Comm. Statist. Theory Methods 37 (18), 2924–2942 (2008)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Durante, F., Klement, E., Quesada-Molina, J.: Bounds for trivariate copulas with given bivariate marginals. J. Inequal. Appl. 2008, 1–9 (2008). Article ID 161537MathSciNetCrossRefGoogle Scholar
  47. 47.
    Durante, F., Klement, E., Quesada-Molina, J., Sarkoci, P.: Remarks on two product-like constructions for copulas. Kybernetika (Prague) 43 (2), 235–244 (2007) 26 Fabrizio Durante and Carlo SempiMATHMathSciNetGoogle Scholar
  48. 48.
    Durante, F., Kolesárová, A., Mesiar, R., Sempi, C.: Copulas with given diagonal sections: novel constructions and applications. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 15 (4), 397–410 (2007)MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    Durante, F., Kolesárová, A., Mesiar, R., Sempi, C.: Copulas with given values on a horizontal and a vertical section. Kybernetika (Prague) 43 (2), 209–220 (2007)MATHMathSciNetGoogle Scholar
  50. 50.
    Durante, F., Quesada-Molina, J.J., Úbeda-Flores, M.: On a family of multivariate copulas for aggregation processes. Inform. Sci. 177 (24), 5715–5724 (2007)MATHMathSciNetCrossRefGoogle Scholar
  51. 51.
    Durante, F., Saminger-Platz, S., Sarkoci, P.: Rectangular patchwork for bivariate copulas and tail dependence. Comm. Statist. Theory Methods 38 (15), 2515–2527 (2009)MATHMathSciNetCrossRefGoogle Scholar
  52. 52.
    Durante, F., Sarkoci, P., Sempi, C.: Shuffles of copulas. J. Math. Anal. Appl. 352 (2), 914–921 (2009)MATHMathSciNetCrossRefGoogle Scholar
  53. 53.
    Durante, F., Sempi, C.: Copula and semicopula transforms. Int. J. Math. Math. Sci. 2005 (4), 645–655 (2005)MATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    Durante, F., Spizzichino, F.: Semi-copulas, capacities and families of level curves. Fuzzy Sets and Systems 161 (2), 269–276 (2009).CrossRefGoogle Scholar
  55. 55.
    Durrleman, V., Nikeghbali, A., Roncalli, T.: A simple transformation of copulas (2000). Available at SSRN: http://ssrn.com/abstract=1032543
  56. 56.
    Embrechts, P.: Discussion of: “Copulas: tales and facts” by T. Mikosch [Extremes 9 (2006), no. 1, 3–20]. Extremes 9 (1), 45–47 (2006)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Embrechts, P.: Copulas: a personal view. J. Risk Ins. 76 (3), 639–650 (2009)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Embrechts, P., McNeil, A.J., Straumann, D.: Correlation and dependence in risk management: properties and pitfalls. In: M. Dempster (ed.) Risk management: value at risk and beyond, pp. 176–223. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  59. 59.
    Embrechts, P., Puccetti, G.: Risk aggregation. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Springer (2010).Google Scholar
  60. 60.
    Erdely, A., González-Barrios, J.M.: On the construction of families of absolutely continuous copulas with given restrictions. Comm. Statist. Theory Methods 35 (4-6), 649–659 (2006)MATHMathSciNetGoogle Scholar
  61. 61.
    Eyraud, H.: Les principes de la mesure des correlations. Ann. Univ. Lyon, III. Ser., Sect. A 1, 30–47 (1936)Google Scholar
  62. 62.
    Fang, H.B., Fang, K.T., Kotz, S.: The meta-elliptical distributions with given marginals. J. Multivariate Anal. 82 (1), 1–16 (2002)MATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Fang, K.T., Kotz, S., Ng, K.W.: Symmetric multivariate and related distributions, Monographs on Statistics and Applied Probability, vol. 36. Chapman and Hall Ltd., London (1990)Google Scholar
  64. 64.
    Farlie, D.J.G.: The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47, 307–323 (1960)MATHMathSciNetGoogle Scholar
  65. 65.
    Féron, R.: Sur les tableaux de corrélation dont les marges sont données. Cas de l’espace a trois dimensions. Publ. Inst. Statist. Univ. Paris 5, 3–12 (1956)MathSciNetGoogle Scholar
  66. 66.
    Fischer, M., Klein, I.: Constructing generalized FGM copulas by means of certain univariate distributions. Metrika 65 (2), 243–260 (2007)MATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Fischer, M., Köck, C., Schlüter, S., Weigert, F.: An empirical analysis of multivariate copula models. Quant. Finance 9 (7), 839–854 (2009)MATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    Frahm, G., Junker, M., Szimayer, A.: Elliptical copulas: applicability and limitations. Statist. Probab. Lett. 63 (3), 275–286 (2003)MATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    Frank, M.J.: On the simultaneous associativity of F(x, y) and x+yF(x, y). Aequationes Math. 19 (2-3), 194–226 (1979)MATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    Fréchet, M.: Sur les tableaux de corrélation dont les marges sont données. Ann. Univ. Lyon. Sect. A. (3) 14, 53–77 (1951)MathSciNetGoogle Scholar
  71. 71.
    Fréchet, M.: Sur les tableaux de corrélation dont les marges sont données. C. R. Acad. Sci. Paris 242, 2426–2428 (1956)MATHMathSciNetGoogle Scholar
  72. 72.
    Fréchet, M.: Remarques au sujet de la note précédente. C. R. Acad. Sci. Paris 246, 2719–2720 (1958)MATHMathSciNetGoogle Scholar
  73. 73.
    Fredricks, G.A., Nelsen, R.B., Rodríguez-Lallena, J.A.: Copulas with fractal supports. Insurance Math. Econom. 37 (1), 42–48 (2005) 1 Copula Theory: an Introduction 27MATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    Frees, E.W., Valdez, E.A.: Understanding relationships using copulas. N. Am. Actuar. J. 2 (1), 1–25 (1998)MATHMathSciNetGoogle Scholar
  75. 75.
    Genest, C.: Preface [International Conference on Dependence Modelling: Statistical Theory and Applications in Finance and Insurance (DeMoSTAFI)]. Canad. J. Statist. 33 (3), 313–314 (2005). Held in Québec City, QC, May 20–22, 2004MathSciNetCrossRefGoogle Scholar
  76. 76.
    Genest, C.: Preface [Special issue: Papers presented at the DeMoSTAFI Conference]. Insurance Math. Econom. 37 (1), 1–2 (2005). Held in Québec, QC, May 20–22, 2004MathSciNetCrossRefGoogle Scholar
  77. 77.
    Genest, C., Favre, A.C.: Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrol. Eng. 12 (4), 347–368 (2007)CrossRefGoogle Scholar
  78. 78.
    Genest, C., Favre, A.C., Béliveau, J., Jacques, C.: Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data. Water Resour. Res. 43, W09,401 (2007). Doi:10.1029/2006WR005275CrossRefGoogle Scholar
  79. 79.
    Genest, C., Gendron, M., Bourdeau-Brien, M.: The advent of copulas in finance. Europ. J. Finance 15 (7–8), 609–618 (2009)CrossRefGoogle Scholar
  80. 80.
    Genest, C., Ghoudi, K., Rivest, L.P.: “Understanding relationships using copulas,” by Edward Frees and Emiliano Valdez, January 1998. N. Am. Actuar. J. 2 (3), 143–149 (1998)MathSciNetGoogle Scholar
  81. 81.
    Genest, C., MacKay, R.J.: Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. Canad. J. Statist. 14 (2), 145–159 (1986)MATHMathSciNetCrossRefGoogle Scholar
  82. 82.
    Genest, C., MacKay, R.J.: The joy of copulas: bivariate distributions with uniform marginals. Amer. Statist. 40 (4), 280–283 (1986)MathSciNetCrossRefGoogle Scholar
  83. 83.
    Genest, C., Nešlehová, J.: A primer on copulas for count data. Astin Bull. 37 (2), 475–515 (2007)MATHMathSciNetCrossRefGoogle Scholar
  84. 84.
    Genest, C., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Sempi, C.: A characterization of quasi-copulas. J. Multivariate Anal. 69 (2), 193–205 (1999)MATHMathSciNetCrossRefGoogle Scholar
  85. 85.
    Genest, C., Rémillard, B.: Discussion of: “Copulas: tales and facts” by T. Mikosch [Extremes 9 (2006), no. 1, 3–20]. Extremes 9 (1), 27–36 (2006)MathSciNetCrossRefGoogle Scholar
  86. 86.
    Genest, C., Rémillard, B., Beaudoin, D.: Goodness-of-fit tests for copulas: a review and a power study. Insurance Math. Econom. 44 (2), 199–213 (2009)MATHMathSciNetCrossRefGoogle Scholar
  87. 87.
    Genest, C., Rivest, L.P.: On the multivariate probability integral transformation. Statist. Probab. Lett. 53 (4), 391–399 (2001)MATHMathSciNetCrossRefGoogle Scholar
  88. 88.
    Giacomini, E., Härdle, W., Spokoiny, V.: Inhomogeneous dependence modeling with timevarying copulae. J. Bus. Econom. Statist. 27 (2), 224–234 (2009)MathSciNetCrossRefGoogle Scholar
  89. 89.
    Gudendorf, G., Segers, J.: Extreme value theory and copulae. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Springer (2010).Google Scholar
  90. 90.
    Gumbel, E.J.: Distributions à plusieurs variables dont les marges sont données. C. R. Acad. Sci. Paris 246, 2717–2719 (1958)MATHMathSciNetGoogle Scholar
  91. 91.
    Gumbel, E.J.: Bivariate exponential distributions. J. Amer. Statist. Assoc. 55, 698–707 (1960)MATHMathSciNetCrossRefGoogle Scholar
  92. 92.
    Gumbel, E.J.: Distributions des valeurs extrêmes en plusieurs dimensions. Publ. Inst. Statist. Univ. Paris 9, 171–173 (1960)MATHMathSciNetGoogle Scholar
  93. 93.
    Gumbel, E.J.: Bivariate logistic distributions. J. Amer. Statist. Assoc. 56, 335–349 (1961)MATHMathSciNetCrossRefGoogle Scholar
  94. 94.
    de Haan, L.: Discussion of: “Copulas: tales and facts” by T. Mikosch [Extremes 9 (2006), no. 1, 3–20]. Extremes 9 (1), 21–22 (2006)MathSciNetCrossRefGoogle Scholar
  95. 95.
    Härdle, W., Okhrin, O.: De copulis non est disputandum. Copulae: an overview. AStA Adv. Stat. Anal. (2010). In pressGoogle Scholar
  96. 96.
    Hoeffding, W.: Maßstabinvariante Korrelationstheorie. Schriften des Mathematischen Instituts und des Instituts für Angewandte Mathematik der Universität Berlin 5 (3), 179–233 (1940). (Reprinted as “Scale-invariant correlation theory" in Fisher, N. I. and Sen, P. K., editors, The Collected Works of Wassily Hoeffding, pages 57–107. Springer, New York. 1994)Google Scholar
  97. 97.
    Hofert, M.: Construction and sampling of nested Archimedean copulas. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Springer (2010).Google Scholar
  98. 98.
    Höffding, W.: Maszstabinvariante Korrelationstheorie. Schr. Math. Inst. u. Inst. Angew. Math. Univ. Berlin 5, 181–233 (1940) 28 Fabrizio Durante and Carlo SempiMathSciNetGoogle Scholar
  99. 99.
    Hougaard, P.: A class of multivariate failure time distributions. Biometrika 73 (3), 671–678 (1986)MATHMathSciNetGoogle Scholar
  100. 100.
    Hutchinson, T.P., Lai, C.D.: Continuous bivariate distributions, emphasising applications. Rumsby Scientific Publishing, Adelaide (1990)MATHGoogle Scholar
  101. 101.
    Ibragimov, R.: Copula-based characterizations for higher order Markov processes. Economet. Theor. 25 (3), 819–846 (2009)MATHCrossRefGoogle Scholar
  102. 102.
    Jaworski, P.: On copulas and their diagonals. Inform. Sci. 179 (17), 2863–2871 (2009)MATHMathSciNetCrossRefGoogle Scholar
  103. 103.
    Jaworski, P., Rychlik, T.: On distributions of order statistics for absolutely continuous copulas with applications to reliability. Kybernetika (Prague) 44 (6), 757–776 (2008)MATHMathSciNetGoogle Scholar
  104. 104.
    Joe, H.: Multivariate models and dependence concepts, Monographs on Statistics and Applied Probability, vol. 73. Chapman & Hall, London (1997)Google Scholar
  105. 105.
    Joe, H.: Discussion of: “Copulas: tales and facts” by T. Mikosch [Extremes 9 (2006), no. 1, 3–20]. Extremes 9 (1), 37–41 (2006)MathSciNetCrossRefGoogle Scholar
  106. 106.
    Kallenberg, O.: Foundations of modern probability, second edn. Probability and its Applications (New York). Springer-Verlag, New York (2002)Google Scholar
  107. 107.
    Kallsen, J., Tankov, P.: Characterization of dependence of multidimensional Lévy processes using Lévy copulas. J. Multivariate Anal. 97 (7), 1551–1572 (2006)MATHMathSciNetCrossRefGoogle Scholar
  108. 108.
    Khoudraji, A.: Contributions à l’étude des copules et à la modélisation des valeurs extrêmes bivariées. Ph.D. thesis, Université de Laval, Québec (Canada) (1995)Google Scholar
  109. 109.
    Kimeldorf, G., Sampson, A.: Uniform representations of bivariate distributions. Comm. Statist. 4 (7), 617–627 (1975)MathSciNetCrossRefGoogle Scholar
  110. 110.
    Klement, E.P., Kolesárová, A.: Intervals of 1-Lipschitz aggregation operators, quasi-copulas, and copulas with given affine section. Monatsh. Math. 152 (2), 151–167 (2007)MATHMathSciNetCrossRefGoogle Scholar
  111. 111.
    Klement, E.P., Kolesárová, A., Mesiar, R., Sempi, C.: Copulas constructed from horizontal sections. Comm. Statist. Theory Methods 36 (13-16), 2901–2911 (2007)MATHMathSciNetCrossRefGoogle Scholar
  112. 112.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular norms, Trends in Logic—Studia Logica Library, vol. 8. Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  113. 113.
    Klement, E.P., Mesiar, R., Pap, E.: Archimax copulas and invariance under transformations. C. R. Math. Acad. Sci. Paris 340 (10), 755–758 (2005)MATHMathSciNetGoogle Scholar
  114. 114.
    Klement, E.P., Mesiar, R., Pap, E.: Transformations of copulas. Kybernetika (Prague) 41 (4), 425–434 (2005)MathSciNetGoogle Scholar
  115. 115.
    Klüppelberg, C., Resnick, S.I.: The Pareto copula, aggregation of risks, and the emperor’s socks. J. Appl. Probab. 45 (1), 67–84 (2008)MATHMathSciNetCrossRefGoogle Scholar
  116. 116.
    Koehler, K.J., Symanowski, J.T.: Constructing multivariate distributions with specific marginal distributions. J. Multivariate Anal. 55 (2), 261–282 (1995)MATHMathSciNetCrossRefGoogle Scholar
  117. 117.
    Kolesárová, A., Mesiar, R., Sempi, C.: Measure-preserving transformations, copulæ and compatibility. Mediterr. J. Math. 5 (3), 325–339 (2008)MATHMathSciNetCrossRefGoogle Scholar
  118. 118.
    Kolev, N., dos Anjos, U., Mendes, B.: Copulas: a review and recent developments. Stoch. Models 22 (4), 617–660 (2006)MATHMathSciNetCrossRefGoogle Scholar
  119. 119.
    Kollo, T.: Preface. J. Statist. Plan. Infer. 139 (11), 3740 (2009)MathSciNetCrossRefGoogle Scholar
  120. 120.
    Lagerås, A.N.: Copulas for Markovian dependence. Bernoulli (2009). In pressGoogle Scholar
  121. 121.
    Li, D.: On default correlation: a copula function approach. Journal of Fixed Income 9, 43–54 (2001)MATHCrossRefGoogle Scholar
  122. 122.
    Li, H.: Duality of he multivariate distributions of Marshall-Olkin type and tail dependence. Comm. Statist. Theory Methods 37 (11-12), 1721–1733 (2008)MATHMathSciNetCrossRefGoogle Scholar
  123. 123.
    Li, H.: Tail dependence comparison of survival Marshall-Olkin copulas. Methodol. Comput. Appl. Probab. 10 (1), 39–54 (2008)MATHMathSciNetCrossRefGoogle Scholar
  124. 124.
    Li, H., Scarsini, M., Shaked, M.: Linkages: a tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals. J. Multivariate Anal. 56 (1), 20–41 (1996)MATHMathSciNetCrossRefGoogle Scholar
  125. 125.
    Liebscher, E.: Construction of asymmetric multivariate copulas. J. Multivariate Anal. 99 (10), 2234–2250 (2008)MATHMathSciNetCrossRefGoogle Scholar
  126. 126.
    Lindner, A.: Discussion of: “Copulas: tales and facts” by T. Mikosch [Extremes 9 (2006), no. 1, 3–20]. Extremes 9 (1), 43–44 (2006) 1 Copula Theory: an Introduction 29MathSciNetCrossRefGoogle Scholar
  127. 127.
    Mai, J.F., Scherer, M.: Efficiently sampling exchangeable Cuadras-Augé copulas in high dimensions. Inform. Sci. 179 (17), 2872–2877 (2009)MATHMathSciNetCrossRefGoogle Scholar
  128. 128.
    Mai, J.F., Scherer, M.: Lévy-Frailty copulas. J. Multivariate Anal. 100 (7), 1567–1585 (2009)MATHMathSciNetCrossRefGoogle Scholar
  129. 129.
    Mai, J.F., Scherer, M.: Reparameterizing Marshall-Olkin copulas with applications to highdimensional sampling. J. Stat. Comput. Simul. (2009) In pressGoogle Scholar
  130. 130.
    Malevergne, Y., Sornette, D.: Extreme financial risks. Springer-Verlag, Berlin (2006)MATHGoogle Scholar
  131. 131.
    Mardia, K.V.: Multivariate Pareto distributions. Ann. Math. Statist. 33, 1008–1015 (1962)MATHMathSciNetCrossRefGoogle Scholar
  132. 132.
    Mardia, K.V.: Families of bivariate distributions. Hafner Publishing Co., Darien, Conn. (1970). Griffin’s Statistical Monographs and Courses, No. 27MATHGoogle Scholar
  133. 133.
    Marshall, A.W.: Copulas, marginals, and joint distributions. In: Distributions with fixed marginals and related topics (Seattle, WA, 1993), IMS Lecture Notes Monogr. Ser., vol. 28, pp. 213–222. Inst. Math. Statist., Hayward, CA (1996)Google Scholar
  134. 134.
    Marshall, A.W., Olkin, I.: A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44 (1967)MATHMathSciNetCrossRefGoogle Scholar
  135. 135.
    Mayor, G., Mesiar, R., Torrens, J.: On quasi-homogeneous copulas. Kybernetika (Prague) 44 (6), 745–756 (2008)MATHMathSciNetGoogle Scholar
  136. 136.
    Mc Neil, A.J., Nešlehová, J.: From Archimedean to Liouville copulas (2009). SubmittedGoogle Scholar
  137. 137.
    Mc Neil, A.J., Nešlehová, J.: Multivariate Archimedean copulas, d-monotone functions and _1-norm symmetric distributions. Ann. Statist. 37 (5B), 3059–3097 (2009)MathSciNetCrossRefGoogle Scholar
  138. 138.
    McNeil, A.J., Frey, R., Embrechts, P.: Quantitative risk management. Concepts, techniques and tools. Princeton Series in Finance. Princeton University Press, Princeton, NJ (2005)Google Scholar
  139. 139.
    Mesiar, R., Sempi, C.: Ordinal sums and idempotents of copulas. Aequationes Math. (2009). In pressGoogle Scholar
  140. 140.
    Mikosch, T.: Copulas: tales and facts. Extremes 9 (1), 3–20 (2006)MathSciNetCrossRefGoogle Scholar
  141. 141.
    Mikosch, T.: “Copulas: tales and facts” [Extremes 9 (2006), no. 1, 3–20]—rejoinder. Extremes 9 (1), 55–62 (2006)MathSciNetCrossRefGoogle Scholar
  142. 142.
    Mikusi´nski, P., , Taylor, M.D.: Some approximations of n-copulas. Metrika (2009). In pressGoogle Scholar
  143. 143.
    Mikusi´nski, P., Sherwood, H., Taylor, M.D.: Probabilistic interpretations of copulas and their convex sums. In: Advances in probability distributions with given marginals (Rome, 1990), Math. Appl., vol. 67, pp. 95–112. Kluwer Acad. Publ., Dordrecht (1991)Google Scholar
  144. 144.
    Mikusi´nski, P., Sherwood, H., Taylor, M.D.: Shuffles of Min. Stochastica 13 (1), 61–74 (1992)MathSciNetGoogle Scholar
  145. 145.
    Moore, D.S., Spruill, M.C.: Unified large-sample theory of general chi-squared statistics for tests of fit. Ann. Statist. 3, 599–616 (1975)MATHMathSciNetGoogle Scholar
  146. 146.
    Morgenstern, D.: Einfache Beispiele zweidimensionaler Verteilungen. Mitteilungsbl. Math. Statist. 8, 234–235 (1956)MathSciNetGoogle Scholar
  147. 147.
    Morillas, P.M.: A method to obtain new copulas from a given one. Metrika 61 (2), 169–184 (2005)MATHMathSciNetCrossRefGoogle Scholar
  148. 148.
    Nadarajah, S.: Marshall and Olkin’s distributions. Acta Appl. Math. 103 (1), 87–100 (2008)MATHMathSciNetCrossRefGoogle Scholar
  149. 149.
    Nappo, G., Spizzichino, F.: Kendall distributions and level sets in bivariate exchangeable survival models. Inform. Sci. 179 (17), 2878–2890 (2009)MATHMathSciNetCrossRefGoogle Scholar
  150. 150.
    Nelsen, R.B.: An introduction to copulas, Lecture Notes in Statistics, vol. 139. Springer- Verlag, New York (1999)Google Scholar
  151. 151.
    Nelsen, R.B.: An introduction to copulas, second edn. Springer Series in Statistics. Springer, New York (2006)Google Scholar
  152. 152.
    Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M.: Bounds on bivariate distribution functions with given margins and measures of association. Comm. Statist. Theory Methods 30 (6), 1155–1162 (2001)MATHMathSciNetCrossRefGoogle Scholar
  153. 153.
    Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M.: Some new properties of quasi-copulas. In: C. Cuadras, J. Fortiana, J. Rodrí guez Lallena (eds.) Distributions with given marginals and Statistical Modelling, pp. 187–194. Kluwer, Dordrecht (2003)Google Scholar
  154. 154.
    Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M.: Bestpossible bounds on sets of bivariate distribution functions. J. Multivariate Anal. 90 (2), 348–358 (2004) 30 Fabrizio Durante and Carlo SempiMATHMathSciNetCrossRefGoogle Scholar
  155. 155.
    Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M.: On the construction of copulas and quasi-copulas with given diagonal sections. Insurance Math. Econom. 42 (2), 473–483 (2008)MATHMathSciNetCrossRefGoogle Scholar
  156. 156.
    Nelsen, R.B., Úbeda-Flores, M.: A comparison of bounds on sets of joint distribution functions derived from various measures of association. Comm. Statist. Theory Methods 33 (10), 2299–2305 (2004)MATHMathSciNetCrossRefGoogle Scholar
  157. 157.
    Nelsen, R.B., Úbeda-Flores, M.: The lattice-theoretic structure of sets of bivariate copulas and quasi-copulas. C. R. Math. Acad. Sci. Paris 341 (9), 583–586 (2005)MATHMathSciNetGoogle Scholar
  158. 158.
    Oakes, D.: A model for association in bivariate survival data. J. Roy. Statist. Soc. Ser. B 44 (3), 414–422 (1982)MATHMathSciNetGoogle Scholar
  159. 159.
    Owzar, K., Sen, P.K.: Copulas: concepts and novel applications. Metron 61 (3), 323–353 (2004) (2003)MathSciNetGoogle Scholar
  160. 160.
    Patton, A.J.: Copula-based models for financial time series. In: T.G. Andersen, R.A. Davis, J.P. Kreiss, T. Mikosch (eds.) Handbook of Financial Time Series, pp. 767–785. Springer (2009)Google Scholar
  161. 161.
    Peng, L.: Discussion of: “Copulas: tales and facts” by T. Mikosch [Extremes 9 (2006), no. 1, 3–20]. Extremes 9 (1), 49–50 (2006)MATHMathSciNetCrossRefGoogle Scholar
  162. 162.
    Puccetti, G., Scarsini, M.: Multivariate comonotonicity. J. Multivariate Anal. 101 (1), 291–304 (2010).MATHMathSciNetCrossRefGoogle Scholar
  163. 163.
    Quesada-Molina, J.J., Rodríguez-Lallena, J.A.: Some advances in the study of the compatibility of three bivariate copulas. J. Ital. Stat. Soc. 3 (3), 397–417 (1994)MATHCrossRefGoogle Scholar
  164. 164.
    Quesada-Molina, J.J., Rodríguez-Lallena, J.A.: Bivariate copulas with quadratic sections. J. Nonparametr. Statist. 5 (4), 323–337 (1995)MATHMathSciNetCrossRefGoogle Scholar
  165. 165.
    Quesada-Molina, J.J., Saminger-Platz, S., Sempi, C.: Quasi-copulas with a given subdiagonal section. Nonlinear Anal. 69 (12), 4654–4673 (2008)MATHMathSciNetCrossRefGoogle Scholar
  166. 166.
    Rényi, A.: On measures of dependence. Acta Math. Acad. Sci. Hungar. 10, 441–451 (1959)MATHMathSciNetCrossRefGoogle Scholar
  167. 167.
    Resnick, S.I.: Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4. Springer-Verlag, New York (1987)Google Scholar
  168. 168.
    Rodríguez-Lallena, J.A.: A class of copulas with piecewise linear horizontal sections. J. Statist. Plan. Infer. 139 (11), 3908–3920 (2009)MATHCrossRefGoogle Scholar
  169. 169.
    Rodríguez-Lallena, J.A., Úbeda-Flores, M.: Best-possible bounds on sets of multivariate distribution functions. Comm. Statist. Theory Methods 33 (4), 805–820 (2004)MATHMathSciNetCrossRefGoogle Scholar
  170. 170.
    Rodríguez-Lallena, J.A., Úbeda-Flores, M.: A new class of bivariate copulas. Statist. Probab. Lett. 66 (3), 315–325 (2004)MATHMathSciNetCrossRefGoogle Scholar
  171. 171.
    Rodríguez-Lallena, J.A., Úbeda-Flores, M.: Multivariate copulas with quadratic sections in one variable. Metrika (2009). In pressGoogle Scholar
  172. 172.
    Rodríguez-Lallena, J.A., Úbeda-Flores, M.: Some new characterizations and properties of quasi–copulas. Fuzzy Sets and Systems 160 (6), 717–725 (2009)MATHMathSciNetCrossRefGoogle Scholar
  173. 173.
    Rüschendorf, L.: Sharpness of Fréchet-bounds. Z. Wahrsch. Verw. Gebiete 57 (2), 293–302 (1981)MATHMathSciNetCrossRefGoogle Scholar
  174. 174.
    Rüschendorf, L.: Construction of multivariate distributions with given marginals. Ann. Inst. Statist. Math. 37 (2), 225–233 (1985)MATHMathSciNetCrossRefGoogle Scholar
  175. 175.
    Rüschendorf, L.: On the distributional transform, Sklar’s Theorem, and the empirical copula process. J. Statist. Plan. Infer. 139 (11), 3921–3927 (2009)MATHCrossRefGoogle Scholar
  176. 176.
    Rüschendorf, L., Schweizer, B., Taylor, M. (eds.): Distributions with fixed marginals and related topics, Institute of Mathematical Statistics Lecture Notes—Monograph Series, 28. Institute of Mathematical Statistics, Hayward, CA (1996)Google Scholar
  177. 177.
    Salvadori, G., De Michele, C., Kottegoda, N.T., Rosso, R.: Extremes in Nature. An Approach Using Copulas, Water Science and Technology Library, vol. 56. Springer, Dordrecht (NL) (2007)Google Scholar
  178. 178.
    Sarmanov, O.V.: Generalized normal correlation and two-dimensional Fréchet classes. Dokl. Akad. Nauk SSSR 168, 32–35 (1966)MATHMathSciNetGoogle Scholar
  179. 179.
    Scarsini, M.: Copulae of probability measures on product spaces. J. Multivariate Anal. 31 (2), 201–219 (1989) 1 Copula Theory: an Introduction 31MATHMathSciNetCrossRefGoogle Scholar
  180. 180.
    Schmid, F.: Copula-based measures of multivariate association. In: Jaworski, P., Durante, F., Härdle, W., Rychlik, T.: (eds.) Copula Theory and Its Applications, Proceedings of the Workshop Held in Warsaw 25-26 September 2009, Springer (2010).Google Scholar
  181. 181.
    Schönbucker, P.: Credit Derivatives Pricing Models: Models, Pricing, Implementation. Wiley Finance Series. John Wiley & Sons Ltd., Chichester (2003)Google Scholar
  182. 182.
    Schweizer, B.: Thirty years of copulas. In: G. Dall’Aglio, S. Kotz, G. Salinetti (eds.) Advances in probability distributions with given marginals (Rome, 1990), Math. Appl., vol. 67, pp. 13–50. Kluwer Acad. Publ., Dordrecht (1991)Google Scholar
  183. 183.
    Schweizer, B.: Introduction to copulas. J. Hydrol. Eng. 12 (4), 346–346 (2007)CrossRefGoogle Scholar
  184. 184.
    Schweizer, B., Sklar, A.: Espaces métriques aléatoires. C. R. Acad. Sci. Paris 247, 2092–2094 (1958)MATHMathSciNetGoogle Scholar
  185. 185.
    Schweizer, B., Sklar, A.: Operations on distribution functions not derivable from operations on random variables. Studia Math. 52, 43–52 (1974)MATHMathSciNetGoogle Scholar
  186. 186.
    Schweizer, B., Sklar, A.: Probabilistic metric spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing Co., New York (1983)Google Scholar
  187. 187.
    Schweizer, B., Wolff, E.F.: Sur une mesure de dépendance pour les variables aléatoires. C. R. Acad. Sci. Paris Sér. A 283, 659–661 (1976)MATHMathSciNetGoogle Scholar
  188. 188.
    Schweizer, B., Wolff, E.F.: On nonparametric measures of dependence for random variables. Ann. Statist. 9 (4), 879–885 (1981)MATHMathSciNetCrossRefGoogle Scholar
  189. 189.
    Segers, J.: Efficient estimation of copula parameter. Discussion of: “Copulas: tales and facts” [Extremes 9 (2006), no. 1, 3–20] by T. Mikosch. Extremes 9 (1), 51–53 (2006)MathSciNetCrossRefGoogle Scholar
  190. 190.
    Sempi, C.: Copulæ and their uses. In: K. Doksum, B. Lindquist (eds.) Mathematical and Statistical Methods in Reliability, pp. 73–86. World Scientific, Singapore (2003)CrossRefGoogle Scholar
  191. 191.
    Siburg, K.F., Stoimenov, P.A.: Gluing copulas. Comm. Statist. Theory Methods 37 (19), 3124–3134 (2008)MathSciNetGoogle Scholar
  192. 192.
    Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231 (1959)MathSciNetGoogle Scholar
  193. 193.
    Sklar, A.: Random variables, joint distribution functions, and copulas. Kybernetika (Prague) 9, 449–460 (1973)MATHMathSciNetGoogle Scholar
  194. 194.
    Sklar, A.: Random variables, distribution functions, and copulas—a personal look backward and forward. In: Distributions with fixed marginals and related topics (Seattle, WA, 1993), IMS Lecture Notes Monogr. Ser., vol. 28, pp. 1–14. Inst. Math. Statist., Hayward, CA (1996)Google Scholar
  195. 195.
    Takahasi, K.: Note on the multivariate Burr’s distribution. Ann. Inst. Statist. Math. 17, 257–260 (1965)MATHMathSciNetCrossRefGoogle Scholar
  196. 196.
    Trivedi, P.K., Zimmer, D.M.: Copula Modeling: An Introduction for Practitioners, vol. 1 (2005)Google Scholar
  197. 197.
    Úbeda-Flores, M.: Multivariate copulas with cubic sections in one variable. J. Nonparametr. Statist. 20 (1), 91–98 (2008)CrossRefGoogle Scholar
  198. 198.
    Whitehouse, M.: How a formula ignited market that burned some big investors. The Wall Street Journal (2005). Published on September 12, 2005Google Scholar
  199. 199.
    Williams, D.: Probability with martingales. Cambridge Mathematical Textbooks. Cambridge University Press, Cambridge (1991)Google Scholar
  200. 200.
    Wolff, E.F.: Measures of dependence derived from copulas. Ph.D. thesis, University of Massachusetts, Amherst (1977)Google Scholar
  201. 201.
    Wolff, E.F.: n-dimensional measures of dependence. Stochastica 4 (3), 175–188 (1980)MATHMathSciNetGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Knowledge-Based Mathematical SystemsJohannes Kepler UniversityLinzAustria
  2. 2.Dipartimento di Matematica “Ennio De Giorgi”Università del SalentoLecceItaly

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