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Modelling the joint distribution of competing risks survival times using copula functions

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  • Kaishev, Vladimir K.
  • Dimitrova, Dimitrina S.
  • Haberman, Steven

Abstract

The problem of modelling the joint distribution of survival times in a competing risks model, using copula functions, is considered. In order to evaluate this joint distribution and the related overall survival function, a system of non-linear differential equations is solved, which relates the crude and net survival functions of the modelled competing risks, through the copula. A similar approach to modelling dependent multiple decrements was applied by Carriere [Carriere, J., 1994. Dependent decrement theory. Transactions, Society of Actuaries XLVI, 45-65] who used a Gaussian copula applied to an incomplete double-decrement model which makes it difficult to calculate any actuarial functions and draw relevant conclusions. Here, we extend this methodology by studying the effect of complete and partial elimination of up to four competing risks on the overall survival function, the life expectancy and life annuity values. We further investigate how different choices of the copula function affect the resulting joint distribution of survival times and in particular the actuarial functions which are of importance in pricing life insurance and annuity products. For illustrative purposes, we have used a real data set and used extrapolation to prepare a complete multiple-decrement model up to age 120. Extensive numerical results illustrate the sensitivity of the model with respect to the choice of copula and its parameter(s).

Suggested Citation

  • Kaishev, Vladimir K. & Dimitrova, Dimitrina S. & Haberman, Steven, 2007. "Modelling the joint distribution of competing risks survival times using copula functions," Insurance: Mathematics and Economics, Elsevier, vol. 41(3), pages 339-361, November.
    • Handle: RePEc:eee:insuma:v:41:y:2007:i:3:p:339-361
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    1. Yicheng Zhou & Zhenzhou Lu & Yan Shi & Kai Cheng, 2019. "The copula-based method for statistical analysis of step-stress accelerated life test with dependent competing failure modes," Journal of Risk and Reliability, , vol. 233(3), pages 401-418, June.
    2. Herbert Hove & Frank Beichelt & Parmod K. Kapur, 2017. "Estimation of the Frank copula model for dependent competing risks in accelerated life testing," International Journal of System Assurance Engineering and Management, Springer;The Society for Reliability, Engineering Quality and Operations Management (SREQOM),India, and Division of Operation and Maintenance, Lulea University of Technology, Sweden, vol. 8(4), pages 673-682, December.
    3. Tie Chen & Songlin Zheng & Jinzhi Feng, 2017. "Statistical dependency analysis of multiple competing failure causes of fuel cell engines," Journal of Risk and Reliability, , vol. 231(2), pages 83-90, April.
    4. Graziani, Rebecca & NIGRI, ANDREA, 2023. "An Age–Period–Cohort Model in a Dirichlet Framework: A Coherent Causes of Death Estimation," SocArXiv 856yw, Center for Open Science.
    5. Ying Jiao & Yahia Salhi & Shihua Wang, 2021. "Dynamic Bivariate Mortality Modelling," Working Papers hal-03244324, HAL.
    6. Dimitrova, Dimitrina S. & Haberman, Steven & Kaishev, Vladimir K., 2013. "Dependent competing risks: Cause elimination and its impact on survival," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 464-477.
    7. Nicholas Bett & Juma Kasozi & Daniel Ruturwa, 2023. "Dependency Modeling Approach of Cause-Related Mortality and Longevity Risks: HIV/AIDS," Risks, MDPI, vol. 11(2), pages 1-18, February.
    8. Quanrui Song & Jianxu Liu & Songsak Sriboonchitta, 2019. "Risk Measurement of Stock Markets in BRICS, G7, and G20: Vine Copulas versus Factor Copulas," Mathematics, MDPI, vol. 7(3), pages 1-16, March.
    9. Romera, Rosario & Molanes, Elisa M., 2008. "Copulas in finance and insurance," DES - Working Papers. Statistics and Econometrics. WS ws086321, Universidad Carlos III de Madrid. Departamento de Estadística.
    10. N. Unnikrishnan Nair & P. G. Sankaran & Preethi John, 2018. "Modelling bivariate lifetime data using copula," METRON, Springer;Sapienza Università di Roma, vol. 76(2), pages 133-153, August.
    11. Li, Han & Li, Hong & Lu, Yang & Panagiotelis, Anastasios, 2019. "A forecast reconciliation approach to cause-of-death mortality modeling," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 122-133.
    12. Ying Jiao & Yahia Salhi & Shihua Wang, 2022. "Dynamic Bivariate Mortality Modelling," Methodology and Computing in Applied Probability, Springer, vol. 24(2), pages 917-938, June.
    13. Nicholas Bett & Juma Kasozi & Daniel Ruturwa, 2022. "Temporal Clustering of the Causes of Death for Mortality Modelling," Risks, MDPI, vol. 10(5), pages 1-34, May.

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