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Keywords:
Archimedean copulas; Cox model; dependence; distorted copulas; ordering
Archimedean copulas; Cox model; dependence; distorted copulas; ordering
Summary:
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.
References:
[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 10.1016/0047-259X(78)90063-5 | MR 0512610
[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 10.1093/biomet/83.1.29 | MR 1399153
[3] Charpentier J., Juri A.: Limiting dependence structures for tail events, with applications to credit derivatives. J. Appl. Probab. 44 (2006), 563–586 DOI 10.1239/jap/1152413742 | MR 2248584 | Zbl 1117.62049
[4] Charpentier A., Segers J.: Lower tail dependence for Archimedean copulas: Characterizations and pitfalls. Insurance Math. Econom. 40 (2007), 525–532 DOI 10.1016/j.insmatheco.2006.08.004 | MR 2311548 | Zbl 1183.62086
[5] Charpentier A., Segers J.: Convergence of Archimedean copulas. Prob. Statist. Lett. 78 (2008), 412–419 DOI 10.1016/j.spl.2007.07.014 | MR 2396414 | Zbl 1139.62303
[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 10.1093/biomet/65.1.141 | MR 0501698 | Zbl 0394.92021
[8] Colangelo A., Scarsini, M., Shaked M.: Some positive dependence stochastic orders. J. Multivariate Anal. 97 (2006), 46–78 DOI 10.1016/j.jmva.2004.11.006 | MR 2208843 | Zbl 1086.62009
[9] Cooper R.: A note on certain inequalities. J. London Math. Soc. 2 (1927), 159–163 DOI 10.1112/jlms/s1-2.3.159 | MR 1574413
[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 1576944
[11] Durante F., Sempi C.: Copula and semicopula transforms. Internat. J. Math. Math. Sci. 4 (2005), 645–655 DOI 10.1155/IJMMS.2005.645 | MR 2172400 | Zbl 1078.62055
[12] Durante F., Foschi, F., Spizzichino F.: Threshold copulas and positive dependence. Statist. Probab. Lett., to appear MR 2474379 | Zbl 1148.62032
[13] Feller W.: An Introduction to Probability Theory and Its Applications. Volume 2. Wiley, New York 1971 MR 0270403 | Zbl 0598.60003
[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 10.1016/j.insmatheco.2005.06.010 | MR 2197302 | Zbl 1112.62011
[15] Genest C.: The joy of copulas: bivariate distributions with uniform marginals. Amer. Statist. 40 (1086), 4, 280–283 MR 0866908
[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 0849869 | Zbl 0605.62049
[17] Genest C., Rivest L. P.: On the multivariate probability integral transformation. Statist. Probab. Lett. 53 (2001), 391–399 DOI 10.1016/S0167-7152(01)00047-5 | MR 1856163 | Zbl 0982.62056
[18] Gumbel E. J.: Bivariate logistic distributions. J. Amer. Statist. Assoc. 56 (1961), 335–349 DOI 10.1080/01621459.1961.10482117 | MR 0158451 | Zbl 0099.14502
[19] Joe H.: Multivariate Models and Dependence Concepts. Chapman&Hall, London 1997 MR 1462613 | Zbl 0990.62517
[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 10.1016/j.jbankfin.2005.05.014
[21] Juri A., Wüthrich M. V.: Copula convergence theorems for tail events. Insurance Math. Econom. 30 (2002), 411–427 DOI 10.1016/S0167-6687(02)00121-X | MR 1921115 | Zbl 1039.62043
[22] Juri A., Wüthrich M. V.: Tail dependence from a distributional point of view. Extremes 6 (2003), 213–246 DOI 10.1023/B:EXTR.0000031180.93684.85 | MR 2081852 | Zbl 1049.62055
[23] Klement E. P., Mesiar, R., Pap E.: Transformations of copulas. Kybernetika 41 (2005), 425–434 MR 2180355
[24] Lehmann E. L.: Some concepts of dependence. Ann. Math. Statist. 37 (1966), 1137–1153 DOI 10.1214/aoms/1177699260 | MR 0202228 | Zbl 0146.40601
[25] Ling C. M.: Representation of associative functions. Publ. Math. Debrecen 12 (1965), 189–212 MR 0190575
[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 10.1007/s001840400330 | MR 2159414 | Zbl 1079.62056
[28] Nelsen R.: An Introduction to Copulas. Springer, New York 1999 MR 1653203 | Zbl 1152.62030
[29] Oakes D.: Bivariate survival models induced by frailties. J. Amer. Statist. Assoc. 84 (1989), 487–493 DOI 10.1080/01621459.1989.10478795 | MR 1010337 | Zbl 0677.62094
[30] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North-Holland, Amsterdam 1959 MR 0790314 | Zbl 0546.60010
[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 0125600
[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 10.1093/biomet/82.1.127 | MR 1332844 | Zbl 0823.62099
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