Experience rating in the classic Markov chain life insurance setting

An empirical Bayes and multivariate frailty approach
  • Christian FurrerEmail author
Original Research Paper


We consider experience rating in the classic Markov chain life insurance setting. We focus on shrinkage estimation of group effects in an empirical Bayes and multivariate frailty extension, building on ideas from group life insurance and survival and event history analysis. Within this framework, we provide insights regarding the structure of the likelihoods and sufficiency of summary statistics such as occurrences and exposures. Simple shrinkage estimators, given by well-known credibility formulas, are obtained under quadratic loss for mutually independent conjugate Gamma priors. The applicability of these simple shrinkage estimators for disability insurance is illustrated in a numerical example using simulated data.


Classic Markov chain life insurance setting Empirical Bayes Experience rating Multivariate frailty Shrinkage 


  1. 1.
    Aalen O, Borgan Ø, Gjessing H (2008) Survival and event history analysis: a process point of view. Statistics for biology and health. Springer, BerlinCrossRefzbMATHGoogle Scholar
  2. 2.
    Andersen P, Borgan Ø, Gill R, Keiding N (1993) Statistical models based on counting processes. Springer series in statistics. Springer, BerlinCrossRefzbMATHGoogle Scholar
  3. 3.
    Bijwaard G (2014) Multistate event history analysis with frailty. Demogr Res 30:1591–1620CrossRefGoogle Scholar
  4. 4.
    Bühlmann H, Straub E (1970) Glaubwürdigkeit für Schadensätze. Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker 70:111–133zbMATHGoogle Scholar
  5. 5.
    Christiansen MC (2012) Multistate models in health insurance. Adv Stat Anal 96(2):155–186MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christiansen MC, Schinzinger E (2016) A credibility approach for combining likelihood of generalized linear models. ASTIN Bull 46(3):531–569MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deprez P, Shevchenko P, Wüthrich M (2017) Machine learning techniques for mortality modeling. Eur Actuar J 7(2):337–352MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gerds TA, Schumacher M (2006) Consistent estimation of the expected brier score in general survival models with right-censored event times. Biom J 48(6):1029–1040MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ghitany ME, Karlis D, Al-Mutairi DK, Al-Awadhi FA (2012) An EM algorithm for multivariate mixed poisson regression models and its application. Appl Math Sci 6(137):6843–6856MathSciNetGoogle Scholar
  10. 10.
    Gschlössl S, Schoenmaekers P, Denuit M (2011) Risk classification in life insurance: methodology and case study. Eur Actuar J 1:23–41MathSciNetCrossRefGoogle Scholar
  11. 11.
    Haastrup S (2000) Comparison of some Bayesian analyses of heterogeneity in group life insurance. Scand Actuar J 2000:2–16MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hoem J (1969) Markov chain models in life insurance. Blätter der DGVFM 9:91–107CrossRefzbMATHGoogle Scholar
  13. 13.
    Hoem J (1972) Inhomogeneous semi-Markov processes, select actuarial tables, and duration-dependency in demography. In: Greville T (ed) Popul Dyn. Academic Press, New York, pp 251–296CrossRefGoogle Scholar
  14. 14.
    Hougaard P (2000) Analysis of multivariate survival data. Statistics for biology and health. Springer, New YorkCrossRefzbMATHGoogle Scholar
  15. 15.
    Jacobsen M (1982) Statistical analysis of counting processes. Lecture notes in statistics. Springer, New YorkCrossRefzbMATHGoogle Scholar
  16. 16.
    Jacobsen M (2006) Point process theory and applications: marked point and piecewise deterministic processes. Probability and its applications. Birkhäuser, BaselzbMATHGoogle Scholar
  17. 17.
    Janssen J (1966) Application des processus semi-markoviens à un problème d’invaliditè. Bulletin de l’Association Royale des Actuaires Belges 63:35–52Google Scholar
  18. 18.
    Jarner S, Møller T (2015) A partial internal model for longevity risk. Scand Actuar J 4:352–382MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Klugman S, Rhodes T, Purushotham M, Gill S (2009) Credibility theory practices. Soc ActuarGoogle Scholar
  20. 20.
    Lee Y, Nelder J (1996) Hierarchical generalized linear models. J R Stat Soc B 58(4):619–678MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lewis PAW, Shedler GS (1979) Simulation of non-homogeneous Poisson processes by thinning. Naval Res Log Q 26(3):403–413CrossRefzbMATHGoogle Scholar
  22. 22.
    Norberg R (1989) A class of conjugate hierarchical priors for Gammoid likelihoods. Scand Actuar J 4:177–193MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Norberg R (1989) Experience rating in group life insurance. Scand Actuar J 4:194–224MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Norberg R (1991) Reserves in life and pension insurance. Scand Actuar J 1991:3–24MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sokol A (2015) Revisiting the forward equations for inhomogeneous semi-Markov processes. Preprint
  26. 26.
    Vaupel J, Manton K, Stallard E (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16(3):439–454CrossRefGoogle Scholar
  27. 27.
    Wong WH (1986) Theory of partial likelihoods. Ann Stat 14:88–123MathSciNetCrossRefGoogle Scholar

Copyright information

© EAJ Association 2019

Authors and Affiliations

  1. 1.University of CopenhagenCopenhagen ØDenmark
  2. 2.PFA PensionCopenhagen ØDenmark

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