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Nonparametric estimation of multivariate distribution function for truncated and censored lifetime data

  • Valery BaskakovEmail author
  • Anna Bartunova
Original Research Paper
  • 51 Downloads

Abstract

A number of models for generating statistical data in various fields of insurance, including life insurance, pensions, and general insurance have been considered. It is shown that the insurance statistics data, as a rule, are truncated and censored, and often multivariate. We propose a non-parametric estimation of the distribution function for multivariate truncated-censored data in the form of a quasi-empirical distribution and a simple iterative algorithm for its construction. To check the accuracy of the proposed evaluation of the distribution function for truncated-censored data, simulation studies have been conducted, which showed its high efficiency. The proposed estimates have been tested for many years by the IAAC Group of Companies in the actuarial valuation of corporate social liabilities according to IAS 19 Employee Benefits. Apart from insurance, some results of the work can be used, for example in medicine, biology, demography, mathematical theory of reliability, etc.

Keywords

Nonparametric estimation Censored and truncated data Multivariate distribution function Survival analysis Iterative algorithm 

Notes

References

  1. 1.
    Akritas MG, van Keilegom I (2003) Estimation of bivariate and marginal distributions with censored data. J R Stat Soc Ser B 65:457–471MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baskakova A, Baskakov V (2014) IBRN reserves estimate on the bases of multivariate censored data of an insurance company. Actuary (Russian) 5:21–25Google Scholar
  3. 3.
    Baskakov V, Baskakov I (2010) On ratemaking and other tasks in non-life insurance. Actuary (Russian) 4:37–41Google Scholar
  4. 4.
    Baskakov V (1996) On an analog of empirical distribution for multivariate censored data. J Math Sci 81(4):2779–2785MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baskakov V, Selivanova A, Gorniakov I (2018) Estimation accuracy improvement under IAS-19. Actuary (Russia) 6:20–26Google Scholar
  6. 6.
    Campbell G (1981) Nonparametric bivariate estimation with randomly censored data. Biometrika 68(2):417–422MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Carriere JF (2000) Bivariate survival models for coupled lives. Scand Actuarial J 1:17–32MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dabrowska DM (1988) Kaplan-Meier estimate on the plane. Ann Stat 16:1475–1489MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dai H, Bao Y (2009) An inverse probability weighted estimator for the bivariate distribution function under right censoring. Stat Prob Lett 79:1789–1797MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dai H, Restaino M, Wang H (2016) A class of nonparametric bivariate survival function estimators for randomly censored and truncated data. Journal of Nonparametric StatisticsGoogle Scholar
  11. 11.
    Dempster A, Laird N, Rubin D (1977) Maximum likelihood estimation from incomplete data. J R Stat Soc, Ser B 39:1–38zbMATHGoogle Scholar
  12. 12.
    Efron B, Petrosian V (1999) Nonparametric methods for doubly truncated data. J Am Stat Assoc 94:824–834MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frees EW, Carriere JF, Valdez EA (1996) Annuity valuation with dependent mortality. J Risk Insur 63(2):229–261CrossRefGoogle Scholar
  14. 14.
    Frydman H (1994) A note on nonparametric estimation of the distribution function from interval-censored and truncated observations. J R Stat Soc, Ser B 56:71–74MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gampe J (2010) Human mortality beyond age 110. In: Maier H, Gampe J, Jeune B, Robine J-M, Vaupel JW, number 7 in demographic research monographs, chapter III, (eds) Supercentenarians. Springer, Heidelberg et al., pp 219–230Google Scholar
  16. 16.
    Gijbels I, Gürler U (1998) Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Statistica Sinica 8:1219–1232MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gürler Ü (1996) Bivariate estimation with right-truncated data. J Am Stat Assoc 91(435):1152–1165MathSciNetzbMATHGoogle Scholar
  18. 18.
    Gürler Ü (1997) Bivariate distributions and hazard functions when a component is randomly truncated. J Multivar Anal 60:20–47MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hougaard P (2001) Analysis of multivariate survival data. Springer, New York, p 452Google Scholar
  20. 20.
    Kaplan EL, Meier P (1958) Nonparametric estimation from incomplate observations. J Am Stat Assoc 53:457–481CrossRefzbMATHGoogle Scholar
  21. 21.
    Klein JP, Moeschberger ML (2003) Survival analysis: techniques for censored and truncated data. Springer, New York, p 536zbMATHGoogle Scholar
  22. 22.
    Lai TL, Ying Z (1991) Estimating a distribution function with truncated and censored data. Ann Stat 19:417–442MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lopez O (2012) A generalization of the kaplan meier estimator for analyzing bivariate mortality under right-censoring and left-truncation with applications in model-checking for survival copula models. Insur, Math Econ 51:505–516MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Luciano E, Spreeuwb J, Vigna E (2008) Modelling stochastic mortality for dependent lives. Insur, Math Econ 43:234–244MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lynden-Bell D (1971) A method of allowing for known observational selection in small samples applied to 3CR quasars, monthly notices. R Astron Soc 155:95–118CrossRefGoogle Scholar
  26. 26.
    Peto R (1973) Experimental survival curves for interval censored data. Appl Stat 22:86–91CrossRefGoogle Scholar
  27. 27.
    Pruitt RC (1993) Identifiability of bivariate survival curves from censored data. J Am Stat Assoc 88(422):573–579MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sankaran PG, Antony AA (2007) Bivariate competing risks models under random left truncation and right censoring. Indian J Stat 69(3):425–447MathSciNetzbMATHGoogle Scholar
  29. 29.
    Shen PS, Yan YF (2008) Nonparametric estimation of the bivariate survival function with left-truncated and right-censored data. J Stat Plan Inference 138:4041–4054MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shen PS (2014) Simple nonparametric estimators of the bivariate survival function under random left truncation and right censoring. Comput Stat 29:641–659MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Turnbull BW (1976) The empirical distribution function with arbitrarily grouped, censored, and truncated data. J R Stat Soc, Ser B 38:290–295MathSciNetzbMATHGoogle Scholar
  32. 32.
    Tsai W-Y, Jewell NP, Wang M-C (1987) A note on the product-limit estimator under right censoring and left truncation. Biometrika 74(4):883–886CrossRefzbMATHGoogle Scholar
  33. 33.
    Tweedie MCK (1984) An index which distinguishes between some important exponential families. In Statistics: Applications and New Directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. (Eds. J. K. Ghosh and J. Roy), Calcutta: Indian Statistical Institute, 579–604Google Scholar
  34. 34.
    Van Der Laan MJ (1996) Efficient estimation in the bivariate censoring model and repairing NPMLE. Ann Stat 24:596–627MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Wang W (2003) Estimating the association parameter for copula models under dependent censoring. J R Stat Soc, Ser B 65:257–273MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Winsch G, Mouchart M, Duchene J (2002) The life table. Modelling survival and death. Kluwer Academic Publishers, NetherlandsGoogle Scholar

Copyright information

© EAJ Association 2019

Authors and Affiliations

  1. 1.International Actuarial Advisory Company (IAAC), Llc Member of European Actuarial & Consulting Services (EURACS)MoscowRussia
  2. 2.RFTAMoscowRussia

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