On Frailties, Archimedean Copulas and Semi-Invariance Under Truncation

  • David OakesEmail author


Definitions and basic properties of bivariate Archimedean copula models for survival data are reviewed with an emphasis on their motivation via frailties. I present some new characterization results for Archimedean copula models based on a notion I call semi-invariance under truncation.


  1. 1.
    Abel, N. H. (1826). Recherche des fonctions de deux quantités variables indépendantes x et y, telles que f(x, y), qui ont la propriétée que f(z, f(x, y)) est une fonction symétrique de z, x et y. Journal fur die Reine und Angewandte Mathematik, 1, 11–15.Google Scholar
  2. 2.
    Aczél, J. (1950). Einige aus funktionalgleichungen zweier veränderlichen ableitbare differentialgleichungen. Acta Scientiarum Mathematicarum (Szeged), 13, 179–189.MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ahmadi Javid, A. (2009). Copulas with truncation-invariance property. Communications in Statistics - Theory and Methods, 38, 3756–3771.MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bandeen-Roche, K. J., & Liang, K. Y. (1996). Modelling failure-time associations in data with multiple levels of clustering. Biometrika, 83, 29–39.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Clayton, D. G. (1978). A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, 141–151.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cox, D. R. (1972). Regression models and life tables (with discussion). Journal of the Royal Statistical Society, Series B: Statistical Methodology, 34, 187–220.zbMATHGoogle Scholar
  7. 7.
    Devroye, L. (2009). On exact simulation algorithms for some distributions related to Jacobi theta functions. Statistics & Probability Letters, 79, 2251–2259.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Feller, W. (1971). An introduction to probability theory and its applications, Vol. II. New Delhi: Wiley.zbMATHGoogle Scholar
  9. 9.
    Frank, M. J. (1979). On the simultaneous associativity of F(x, y)and x + y − F(x, y). Aequationes Mathematicae, 19, 194–226.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Genest, C. (1987). Frank’s family of bivariate distributions. Biometrika, 74, 549–555.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Genest, C., & MacKay, R. J. (1986). Copules Archimédiennes et familles de lois bidimensionnelles dont les marges sont données. The Canadian Journal of Statistics, 14, 145–159.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Genest, C., & Nešlehová, J. (2007). A primer on copulas for count data. Astrophysical Bulletin, 37, 475–515.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gumbel, E. J. (1960). Bivariate exponential distributions. Journal of the American Statistical Association, 55, 698–707.MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hougaard, P. (1986). A class of multivariate failure time distributions. Biometrika, 73, 671–678.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lee, L. (1979). Multivariate distributions having Weibull properties. Journal of Multivariate Analysis, 9, 267–277.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Manatunga, A. K., & Oakes, D. (1996). A measure of association for bivariate frailty distributions. Journal of Multivariate Analysis, 56, 60–74.MathSciNetCrossRefGoogle Scholar
  17. 17.
    McCullagh, P., & Nelder, J. A. (1989). Generalized linear models, 2nd edn. London: Chapman & Hall/CRC.CrossRefGoogle Scholar
  18. 18.
    McNeil, A. J., & Nešlehová, J. (2009). Multivariate Archimedean copulas, d-monotone functions and 1-norm symmetric distributions. The Annals of Statistics, 37, 3059–3097.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Nelsen, R. B. (2006). An introduction to copulas, 2nd edn. New York: Springer.zbMATHGoogle Scholar
  20. 20.
    Oakes, D. (1982). A model for association in bivariate survival data. Journal of the Royal Statistical Society, Series B: Statistical Methodology, 44, 414–422.MathSciNetzbMATHGoogle Scholar
  21. 21.
    Oakes, D. (1989). Bivariate survival models induced by frailties. Journal of the American Statistical Association, 84, 487–493.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Oakes, D. (2005). On the preservation of copula structure under truncation. The Canadian Journal of Statistics, 33, 465–468.MathSciNetCrossRefGoogle Scholar
  23. 23.
    Piaggio, H. T. H. (1962). An elementary treatise on differential equations and their applications. London: G. Bell & Sons.zbMATHGoogle Scholar
  24. 24.
    Scheffé, H. (1959). The analysis of variance. New York: Wiley.zbMATHGoogle Scholar
  25. 25.
    Wang, A., & Oakes, D. (2008). Some properties of the Kendall distribution in bivariate Archimedean copula models under censoring. Statistics & Probability Letters, 78, 2578–2583.MathSciNetCrossRefGoogle Scholar
  26. 26.
    Williams, E. J. (1977). Some representations of stable random variables as products. Biometrika, 64, 167–169.MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zolotarev, V. M. (1966). On representation of stable laws by integrals. Translation in Mathematical Statistics, 6, 84–88.zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Biostatistics and Computational BiologyUniversity of Rochester Medical CenterRochesterUSA

Personalised recommendations