Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula

Statistical Papers volume 60, pages1101–1118(2019)Cite this article

Abstract

The first part of this paper reviews the properties of bivariate dependence measures (Spearman’s rho, Kendall’s tau, Kochar and Gupta’s dependence measure, and Blest’s coefficient) under the generalized Farlie–Gumbel–Morgenstern (FGM) copula. We give a few remarks on the relationship among the bivariate dependence measures, derive Blest’s coefficient, and suggest simplifying the previously obtained expression of Kochar and Gupta’s dependence measure. The second part of this paper derives some useful measures for analyzing bivariate competing risks models under the generalized FGM copula. We obtain the expression of sub-distribution functions under the generalized FGM copula, which has not been discussed in the literature. With the Burr III margins, we show that our expression has a closed form and generalizes the reliability measure previously obtained by Domma and Giordano (Stat Pap 54(3):807–826, 2013).

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References

  1. Amblard C, Girard S (2009) A new extension of bivariate FGM copulas. Metrika 70(1):1–17

    MathSciNet  Article  MATH  Google Scholar 

  2. Amini M, Jabbari H, Mohtashami Borzadaran GR (2011) Aspects of dependence in generalized Farlie-Gumbel-Morgenstern distributions. Commun Stat Simul Comput 40(8):1192–1205

    MathSciNet  Article  MATH  Google Scholar 

  3. Bairamov I, Kotz S (2002) Dependence structure and symmetry of Huang-Kotz FGM distributions and their extensions. Metrika 56(1):55–72

    MathSciNet  Article  MATH  Google Scholar 

  4. Bakoyannis G, Touloumi G (2012) Practical methods for competing risks data: a review. Stat Methods Med Res 21(3):257–272

    MathSciNet  Article  MATH  Google Scholar 

  5. Basu AP, Ghosh JK (1978) Identifiability of the multinormal and other distributions under competing risks model. J Multivar Anal 8(3):413–429

    MathSciNet  Article  MATH  Google Scholar 

  6. Blest DC (2000) Rank correlation-an alternative measure. Aust N Z J Stat 42(1):101–111

    MathSciNet  Article  MATH  Google Scholar 

  7. Braekers R, Veraverbeke N (2005) A copula-graphic estimator for the conditional survival function under dependent censoring. Can J Stat 33(3):429–447

    MathSciNet  Article  MATH  Google Scholar 

  8. Burr IW (1942) Cumulative frequency functions. Ann Math. Stat 13(2):215–232

    MathSciNet  MATH  Google Scholar 

  9. Capéraà P, Genest C (1993) Spearman’s rho is larger than Kendall’s tau for positively dependent random variables. J Nonparametric Stat 2(2):183–194

    Article  MATH  Google Scholar 

  10. Clayton DG (1978) A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65(1):141–151

    MathSciNet  Article  MATH  Google Scholar 

  11. Crowder MJ (2001) Classical competing risks. Chapman and Hall/CRC, Boca Raton

    Google Scholar 

  12. De Uña-Álvarez J, Veraverbeke N (2013) Generalized copula-graphic estimator. Test 22(2):343–360

    MathSciNet  Article  MATH  Google Scholar 

  13. De Uña-Álvarez J, Veraverbeke N (2014) Generalized copula-graphic estimator with left-truncated and right-censored data. Discuss Pap Stat Oper Res. http://jacobo.webs.uvigo.es/presentation_1.pdf

  14. Domma F, Giordano S (2013) A copula-based approach to account for dependence in stress-strength models. Stat Pap 54(3):807–826

    MathSciNet  Article  MATH  Google Scholar 

  15. Domma F, Giordano S (2016) Concomitants of m-generalized order statistics from generalized Farlie-Gumbel-Morgenstern distribution family. J Comput Appl Math 294:413–435

    MathSciNet  Article  MATH  Google Scholar 

  16. Emura T, Chen YH (2016) Gene selection for survival data under dependent censoring: a copula-based approach. Stat Methods Med Res 25(6):2840–2857

    MathSciNet  Article  Google Scholar 

  17. Emura T, Michimae H (2017) A copula-based inference to piecewise exponential models under dependent censoring, with application to time to metamorphosis of salamander larvae. Environ Ecol Stat (in press)

  18. Emura T, Nakatochi M, Murotani K, Rondeau V (2015) A joint frailty-copula model between tumor progression and death for meta-analysis. Stat Methods Med Res. doi:10.1177/0962280215604510

    Google Scholar 

  19. Emura T, Nakatochi M, Matsui S, Michimae H, Rondeau V (2017) Personalized dynamic prediction of death according to tumour progression and high-dimensional genetic factors: meta-analysis with a joint model. Stat Methods Med Res (in press)

  20. Esary JD, Proschan F, Walkup DW (1967) Association of random variables, with applications. Ann Math Stat 38(5):1466–1474

    MathSciNet  Article  MATH  Google Scholar 

  21. Escarela G, Carrière JF (2003) Fitting competing risks with an assumed copula. Stat Methods Med Res 12(4):333–349

    MathSciNet  Article  MATH  Google Scholar 

  22. Eyraud H (1936) Les principes de la mesure des corrélations. Ann Univ Lyon III Ser Sect A 1:30–47

    MATH  Google Scholar 

  23. Farlie DJG (1960) The performance of some correlation coefficients for a general bivariate distribution. Biometrika 47(3–4):307–323

    MathSciNet  Article  MATH  Google Scholar 

  24. Frank MJ (1979) On the simultaneous associativity of \(F(x, y)\) and \(x+y-F(x, y)\). Aequ Math 19:194–226

    MathSciNet  Article  MATH  Google Scholar 

  25. Genest C (1987) Frank’s family of bivariate distributions. Biometrika 74(3):549–555

    MathSciNet  Article  MATH  Google Scholar 

  26. Genest C, Plante JF (2003) On Blest’s measure of rank correlation. Can J Stat 31(1):35–52

    MathSciNet  Article  MATH  Google Scholar 

  27. Gini C (1912) Variabilità e Mutuabilità. Contributo alloStudio delle Distribuzioni e delle Relazioni Statistiche. C, Cuppini, Bologna

    Google Scholar 

  28. Gray RJ (1988) A class of K-sample tests for comparing the cumulative incidence of a competing risk. Ann Stat 16(3):1141–1154

    MathSciNet  Article  MATH  Google Scholar 

  29. Gumbel EJ (1960a) Distributions des valeurs extrêmes en plusiers dimensions. Publications de l’Institut de Statistique de L’Université de Paris 9:171–173

    MATH  Google Scholar 

  30. Gumbel EJ (1960b) Bivariate exponential distributions. J Am Stat Assoc 55(292):698–707

    MathSciNet  Article  MATH  Google Scholar 

  31. Joe H (1993) Parametric families of multivariate distributions with given margins. J Multiv Anal 46(2):262–282

    MathSciNet  Article  MATH  Google Scholar 

  32. Kochar SC, Gupta RP (1987) Competitors of Kendall-tau test for testing independence against positive quadrant dependence. Biometrika 74(3):664–666

    MathSciNet  Article  MATH  Google Scholar 

  33. Lawless JF (2003) Statistical models and methods for lifetime data, 2nd edn. Wiley, Hoboken

    Google Scholar 

  34. Lehmann EL (1966) Some concepts of dependence. Ann Math Stat 37(5):1137–1153

    MathSciNet  Article  MATH  Google Scholar 

  35. Lindsay SR, Wood GR, Woollons RC (1996) Modelling the diameter distribution of forest stands using the Burr distribution. J Appl Stat 23(6):609–620

    Article  Google Scholar 

  36. Louzada F, Suzuki AK, Cancho VG (2013) The FGM long-term bivariate survival copula model: modeling, Bayesian estimation, and case influence diagnostics. Commun Stat Theory Methods 42(4):673–691

    MathSciNet  Article  MATH  Google Scholar 

  37. Lo SMS, Wilke RA (2010) A copula model for dependent competing risks. J R Stat Soc Ser C Appl Stat 59(2):359–376

    MathSciNet  Article  Google Scholar 

  38. Moeschberger ML (1974) Life tests under dependent competing causes of failure. Technometrics 16(1):39–47

    MathSciNet  Article  MATH  Google Scholar 

  39. Morgenstern D (1956) Einfache Beispiele zweidimensionaler Verteilungen. Mitt Math Stat 8:234–235

    MathSciNet  MATH  Google Scholar 

  40. Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York

    Google Scholar 

  41. Nešlehová J (2007) On rank correlation measures for non-continuous random variables. J Multivar Anal 98(3):544–567

    MathSciNet  Article  MATH  Google Scholar 

  42. Rodíguez-Lallena JA, Úbeda-Flores M (2004) A new class of bivariate copulas. Stat Probab Lett 66(3):315–325

    MathSciNet  Article  MATH  Google Scholar 

  43. Scarsini M (1984) On measures of concordance. Stochastica 8(3):201–218

    MathSciNet  MATH  Google Scholar 

  44. Schucany WR, Parr WC, Boyer JE (1978) Correlation structure in Farlie-Gumbel-Morgenstern distributions. Biometrika 65(3):650–653

    MathSciNet  Article  MATH  Google Scholar 

  45. Schweizer B, Wolff EF (1981) On nonparametric measures of dependence for random variables. Ann Stat 9(4):879–885

    MathSciNet  Article  MATH  Google Scholar 

  46. Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statistique de L’Université de Paris 8:229–231

    MATH  Google Scholar 

  47. Therneau T, Lumley T (2016) R survival package, version 2.40-1. Date 2016-10-30

  48. Zheng M, Klein JP (1995) Estimates of marginal survival for dependent competing risks based on an assumed copula. Biometrika 82(1):127–138

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The authors thank the Editor and two anonymous reviewers for their helpful comments that improved the paper. The authors are also thankful to Prof. Christian Genest for his comments on the earlier version of the paper. This work is supported by the research grant funded by Ministry of Science and Technology, Taiwan (MOST 103-2118-M-008-MY2 and 105-2118-M-008-001-MY2).

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  1. Graduate Institute of Statistics, National Central University, No. 300, Zhongda Rd., Zhongli District, Taoyuan City, 32001, Taiwan

    Jia-Han Shih & Takeshi Emura

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  1. Jia-Han Shih
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  2. Takeshi Emura
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Correspondence to Takeshi Emura.

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Shih, J., Emura, T. Bivariate dependence measures and bivariate competing risks models under the generalized FGM copula. Stat Papers 60, 1101–1118 (2019). https://doi.org/10.1007/s00362-016-0865-5

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Keywords

  • Blest’s coefficient
  • Competing risk
  • FGM copula
  • Kendall’s tau
  • Spearman’s rho