Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników


2013 | 1 | 1-36

Tytuł artykułu

On certain transformations of Archimedean copulas: Application to the non-parametric estimation of their generators

Treść / Zawartość

Warianty tytułu

Języki publikacji



We study the impact of certain transformations within the class of Archimedean copulas. We give some admissibility conditions for these transformations, and define some equivalence classes for both transformations and generators of Archimedean copulas. We extend the r-fold composition of the diagonal section of a copula, from r ∈ N to r ∈ R. This extension, coupled with results on equivalence classes, gives us new expressions of transformations and generators. Estimators deriving directly from these expressions are proposed and their convergence is investigated. We provide confidence bands for the estimated generators. Numerical illustrations show the empirical performance of these estimators.








Opis fizyczny




  • Conservatoire National des Arts et Métiers, Département IMATH,
    EA4629, 292 rue Saint Martin, 75011, Paris, France
  • Université de Lyon, Université Lyon 1, ISFA, Laboratoire SAF,
    EA2429, 50 avenue Tony Garnier, 69366 Lyon, France


  • [1] Alsina, C., Schweizer, B., and Frank, M. J. (2006). Associative functions: triangular norms and copulas. World Scientific.
  • [2] Autin, F., Le Pennec, E., and Tribouley, K. (2010). Thresholding methods to estimate copula density. J. Multivariate Anal., 101(1):200–222.
  • [3] Bienvenüe, A. and Rullière, D. (2011). Iterative adjustment of survival functions by composed probability distortions. Geneve Risk Ins. Rev., 37(2):156–179.
  • [4] Bienvenüe, A. and Rullière, D. (2012). On hyperbolic iterated distortions for the adjustment of survival functions. In Perna, C. and Sibillo, M., editors, Mathematical and Statistical Methods for Actuarial Sciences and Finance, pages 35–42. Springer Milan.
  • [5] Brechmann, E. (2013). Sampling from Hierarchical Kendall Copulas. J. SFdS, 154(1):192–209.
  • [6] Charpentier, A. and Segers, J. (2007). Lower tail dependence for Archimedean copulas: characterizations and pitfalls. Insurance Math. Econom., 40(3):525–532.
  • [7] Chomette, T. (2003). Arbres et dérivée d’une fonction composée. draft paper, ENS, http: // www. math. ens. fr/ culturemath/ maths/ pdf/ analyse/ derivation. pdf .
  • [8] Crane, G. and van der Hoek, J. (2008). Using distortions of copulas to price synthetic CDOs. Insurance Math. Econom., 42(3):903 – 908.
  • [9] Curtright, T. and Zachos, C. (2009). Evolution profiles and functional equations. J. Phys. A., 42(48):485208.
  • [10] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. Acad. Roy. Belg. Bull. Cl. Sci., 65(5):274–292.
  • [11] Deheuvels, P. (1980). Non parametric tests of independence. In Raoult, J.-P., editor, Statistique non Paramétrique Asymptotique, volume 821 of Lecture Notes in Math., pages 95–107. Springer Berlin Heidelberg.
  • [12] Di Bernardino, E. and Rullière, D. (2013). Distortions of multivariate distribution functions and associated level curves: Applications in multivariate risk theory. Insurance Math. Econom., 53(1):190 – 205.
  • [13] Durante, F., Foschi, R., and Sarkoci, P. (2010). Distorted copulas: Constructions and tail dependence. Comm. Statist. Theory Methods, 39(12):2288–2301. [Crossref]
  • [14] Durante, F. and Jaworski, P. (2008). Absolutely continuous copulas with given diagonal sections. Comm. Statist. Theory Methods, 37(18):2924–2942. [Crossref]
  • [15] Durante, F. and Sempi, C. (2005). Copula and semicopula transforms. Int. J. Math. Math. Sci., 2005(4):645–655.
  • [16] Durrleman, V., Nikeghbali, A., and Roncalli, T. (2000). A simple transformation of copulas. Technical report, Groupe de Research Operationnelle Credit Lyonnais.
  • [17] Embrechts, P. and Hofert, M. (2011). Comments on: Inference in multivariate Archimedean copula models. TEST, 20(2):263–270. [Crossref]
  • [18] Embrechts, P. and Hofert, M. (2013). Statistical inference for copulas in high dimensions: A simulation study. ASTIN Bull., 43:81–95.
  • [19] Erdely, A., González-Barrios, J. M., and Hernández-Cedillo, M. M. (2013). Frank’s condition for multivariate Archimedean copulas. Fuzzy Sets and Systems, In press(Available online).
  • [20] Fermanian, J.-D., Radulovic, D., andWegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli, 10(5):847–860. [Crossref]
  • [21] Fischer, M. and Köck, C. (2012). Constructing and generalizing given multivariate copulas: A unifying approach. Statistics, 46(1):1–12. [Crossref]
  • [22] Frees, E. W. and Valdez, E. A. (1998). Understanding relationships using copulas. N. Am. Actuar. J., 2(1):1–25.
  • [23] Genest, C., Ghoudi, K., and Rivest, L.-P. (1998). Discussion of “understanding relationships using copulas,” by edward frees and emiliano valdez, january 1998. N. Am. Actuar. J., 2(3):143–149.
  • [24] Genest, C., Masiello, E., and Tribouley, K. (2009). Estimating copula densities through wavelets. Insurance Math. Econom., 44(2):170–181.
  • [25] Genest, C., Nešlehová, J., and Ziegel, J. (2011). Inference in multivariate Archimedean copula models. TEST, 20(2):223–256. [Crossref]
  • [26] Genest, C. and Rivest, L.-P. (1993). Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc., 88(423):1034–1043. [Crossref]
  • [27] Hardy, M. (2006). Combinatorics of partial derivatives. Electron. J. Combin., 13(1)
  • [28] Hernández-Lobato, J. M. and Suárez, A. (2011). Semiparametric bivariate Archimedean copulas. Comput. Statist. Data Anal., 55(6):2038–2058. [Crossref]
  • [29] Hofert, M. (2008). Sampling Archimedean copulas. Comput. Statist. Data Anal., 52(12):5163 – 5174. [Crossref]
  • [30] Hofert, M. (2011). Efficiently sampling nested Archimedean copulas. Comput. Statist. Data Anal., 55(1):57 – 70. [Crossref]
  • [31] Hofert, M., Mächler, M., and McNeil, A. J. (2012). Likelihood inference for Archimedean copulas in high dimensions under known margins. J. Multivariate Anal., 110(0):133 – 150.
  • [32] Hofert, M. and Pham, D. (2013). Densities of nested Archimedean copulas. J. Multivariate Anal., 118(0):37 – 52.
  • [33] Jaworski, P. (2009). On copulas and their diagonals. Inform. Sci., 179(17):2863 – 2871. [Crossref]
  • [34] Jaworski, P. and Rychlik, T. (2008). On distributions of order statistics for absolutely continuous copulas with applications to reliability. Kybernetika (Prague), 44(6):757–776.
  • [35] Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. J. Multivariate Anal., 94(2):401–419.
  • [36] Juri, A. and Wüthrich, M. V. (2002). Copula convergence theorems for tail events. Insurance Math. Econom., 30(3):405–420.
  • [37] Juri, A. and Wüthrich, M. V. (2003). Tail dependence from a distributional point of view. Extremes, 6(3):213–246. [Crossref]
  • [38] Kim, G., Silvapulle, M. J., and Silvapulle, P. (2007). Comparison of semiparametric and parametric methods for estimating copulas. Comput. Statist. Data Anal., 51(6):2836–2850. [Crossref]
  • [39] Klement, E. P., Mesiar, R., and Pap, E. (2005a). Archimax copulas and invariance under transformations. C. R. Math. Acad. Sci. Paris, 340(10):755 – 758.
  • [40] Klement, E. P., Mesiar, R., and Pap, E. (2005b). Transformations of copulas. Kybernetika (Prague), 41(4):425 –434.
  • [41] Kojadinovic, I. and Yan, J. (2010). Modeling multivariate distributions with continuous margins using the copula r package. J. Stat. Softw., 34(9):1–20.
  • [42] Lambert, P. (2007). Archimedean copula estimation using Bayesian splines smoothing techniques. Comput. Statist. Data Anal., 51(12):6307 – 6320. [Crossref]
  • [43] Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika, 83(1):169–187. [Crossref]
  • [44] McNeil, A. and Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and l1−norm symmetric distributions. Ann. Statist., 37(5B):3059–3097. [Crossref]
  • [45] Michiels, F. and De Schepper, A. (2012). How to improve the fit of Archimedean copulas by means of transforms. Statist. Papers, 53(2):345–355. [Crossref]
  • [46] Morillas, P. M. (2005). A method to obtain new copulas from a given one. Metrika, 61(2):169–184. [Crossref]
  • [47] Nelsen, R. B. (1999). An introduction to copulas, volume 139 of Lecture Notes in Statistics. Springer-Verlag, New York.
  • [48] Nelsen, R. B. and Fredricks, G. A. (1997). Diagonal copulas. In Beneš, V. and Štepán, J., editors, Distributions with given Marginals and Moment Problems, pages 121–128. Springer Netherlands.
  • [49] Nelsen, R. B., Quesada-Molina, J., Rodriguez-Lallena, J., and Úbeda-Flores, M. (2009). Kendall distribution functions and associative copulas. Fuzzy Sets and Systems, 160(1):52–57.
  • [50] Nelsen, R. B., Quesada-Molina, J. J., Rodríguez-Lallena, J. A., and Úbeda-Flores, M. (2008). On the construction of copulas and quasi-copulas with given diagonal sections. Insurance Math. Econom., 42(2):473–483.
  • [51] Omelka, M., Gijbels, I., and Veraverbeke, N. (2009). Improved kernel estimation of copulas: weak convergence and goodness-of-fit testing. Ann. Statist., 37(5B):3023–3058. [Crossref]
  • [52] Qu, L. and Yin, W. (2012). Copula density estimation by total variation penalized likelihood with linear equality constraints. Comput. Statist. Data Anal., 56(2):384 – 398. [Crossref]
  • [53] Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Statist., 4:912–923. [Crossref]
  • [54] Segers, J. (2011). Diagonal sections of bivariate Archimedean copulas. Discussion of “Inference in multivariate Archimedean copula models” by Christian Genest, Johanna Nešlehová, and Johanna Ziegel. TEST, 20:281–283. [Crossref]
  • [55] Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli, 18(3):764–782. [Crossref]
  • [56] Valdez, E. and Xiao, Y. (2011). On the distortion of a copula and its margins. Scand. Actuar. J., 4:292–317. [Crossref]
  • [57] Wysocki, W. (2012). Constructing Archimedean copulas from diagonal sections. Statist. Probab. Lett., 82(4):818 – 826.[Crossref]

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.