Dynamic dependence ordering for Archimedean copulas and distorted copulas

Charpentier, Arthur

Kybernetika, Tome 44 (2008) no. 6, pp. 777-794

Abstract (VO)

This paper proposes a general framework to compare the strength of the dependence in survival models, as time changes, i.\,e. given remaining lifetimes $\boldsymbol{X}$, to compare the dependence of $\boldsymbol{X}$ given $\boldsymbol{X}>t$, and $\boldsymbol{X}$ given $\boldsymbol{X}>s$, where $s>t$. More precisely, analytical results will be obtained in the case the survival copula of $\boldsymbol{X}$ is either Archimedean or a distorted copula. The case of a frailty based model will also be discussed in details.

Détail

Classification : 60A05, 60E15, 62H05, 62N01, 62N99, 91B30
Keywords: Archimedean copulas; Cox model; dependence; distorted copulas; ordering

Bibliographie

[1] Ali M., Mikhail, N., Haq N. S.: A class of bivariate distribution including the bivariate logistic given margins. J. Multivariate Anal. 8 (1978), 405–412 | DOI | MR

[2] Bandeen-Roche K. J., Liang K. Y.: Modeling failure-time associations in data with multiple levels of clustering. Biometrika 83 (1996), 29–39 | DOI | MR

[3] Charpentier J., Juri A.: Limiting dependence structures for tail events, with applications to credit derivatives. J. Appl. Probab. 44 (2006), 563–586 | DOI | MR | Zbl

[4] Charpentier A., Segers J.: Lower tail dependence for Archimedean copulas: Characterizations and pitfalls. Insurance Math. Econom. 40 (2007), 525–532 | DOI | MR | Zbl

[5] Charpentier A., Segers J.: Convergence of Archimedean copulas. Prob. Statist. Lett. 78 (2008), 412–419 | DOI | MR | Zbl

[6] Charpentier A., Segers J.: Tails of Archimedean copulas. Submitted

[7] Clayton D. G.: A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65 (1978), 141–151 | DOI | MR | Zbl

[8] Colangelo A., Scarsini, M., Shaked M.: Some positive dependence stochastic orders. J. Multivariate Anal. 97 (2006), 46–78 | DOI | MR | Zbl

[9] Cooper R.: A note on certain inequalities. J. London Math. Soc. 2 (1927), 159–163 | DOI | MR

[10] Cooper R.: The converse of the Cauchy–Holder inequality and the solutions of the inequality $g(x + y) \le g(x) + g(y)$. Proc. London Math. Soc. 2 (1927), 415–432 | MR

[11] Durante F., Sempi C.: Copula and semicopula transforms. Internat. J. Math. Math. Sci. 4 (2005), 645–655 | DOI | MR | Zbl

[12] Durante F., Foschi, F., Spizzichino F.: Threshold copulas and positive dependence. Statist. Probab. Lett., to appear | MR | Zbl

[13] Feller W.: An Introduction to Probability Theory and Its Applications. Volume 2. Wiley, New York 1971 | MR | Zbl

[14] Geluk J. L., Vries C. G. de: Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities. Insurance Math. Econom. 38 (2006), 39–56 | DOI | MR | Zbl

[15] Genest C.: The joy of copulas: bivariate distributions with uniform marginals. Amer. Statist. 40 (1086), 4, 280–283 | MR

[16] Genest C., MacKay R. J.: Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. La revue canadienne de statistique 14 (1986), 145–159 | MR | Zbl

[17] Genest C., Rivest L. P.: On the multivariate probability integral transformation. Statist. Probab. Lett. 53 (2001), 391–399 | DOI | MR | Zbl

[18] Gumbel E. J.: Bivariate logistic distributions. J. Amer. Statist. Assoc. 56 (1961), 335–349 | DOI | MR | Zbl

[19] Joe H.: Multivariate Models and Dependence Concepts. Chapman&Hall, London 1997 | MR | Zbl

[20] Junker M., Szimayer, S., Wagner N.: Nonlinear term structure dependence: Copula functions, empirics, and risk implications. J. Banking & Finance 30 (2006), 1171–1199 | DOI

[21] Juri A., Wüthrich M. V.: Copula convergence theorems for tail events. Insurance Math. Econom. 30 (2002), 411–427 | DOI | MR | Zbl

[22] Juri A., Wüthrich M. V.: Tail dependence from a distributional point of view. Extremes 6 (2003), 213–246 | DOI | MR | Zbl

[23] Klement E. P., Mesiar, R., Pap E.: Transformations of copulas. Kybernetika 41 (2005), 425–434 | MR

[24] Lehmann E. L.: Some concepts of dependence. Ann. Math. Statist. 37 (1966), 1137–1153 | DOI | MR | Zbl

[25] Ling C. M.: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189–212 | MR

[26] McNeil A. J., Neslehova J.: Multivariate Archimedean copulas, D-monotone functions and L1-norm symmetric distributions. Ann. Statist. To appear

[27] Morillas P. M.: A method to obtain new copulas from a given one. Metrika 61 (2005), 169–184 | DOI | MR | Zbl

[28] Nelsen R.: An Introduction to Copulas. Springer, New York 1999 | MR | Zbl

[29] Oakes D.: Bivariate survival models induced by frailties. J. Amer. Statist. Assoc. 84 (1989), 487–493 | DOI | MR | Zbl

[30] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North-Holland, Amsterdam 1959 | MR | Zbl

[31] Sklar A.: Fonctions de répartition à $n$ dimensions et leurs marges. Publ. de l’Institut de Statistique de l’Université de Paris 8 (1959), 229–231 | MR

[32] Wang S., Nelsen, R., Valdez E. A.: Distortion of multivariate distributions: adjustment for uncertainty in aggregating risks. Mimeo 2005

[33] Zheng M., Klein J. P.: Estimates of marginal survival for dependent competing risks based on an assumed copula. Biometrika 82 (1995), 127–138 | DOI | MR | Zbl