European Actuarial Journal

, Volume 7, Issue 1, pp 231–255 | Cite as

A review of Bayesian asymptotics in general insurance applications

Original Research Paper
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Abstract

Over the last two decades, Bayesian methods have been widely used in general insurance applications, ranging from credibility theory to loss-reserves estimation, but this literature rarely addresses questions about the method’s asymptotic properties. In this paper, we review the Bayesian’s notion of posterior consistency in both parametric and nonparametric models and its implication on the sensitivity of the posterior to the actuary’s choice of prior. We review some of the techniques for proving posterior consistency and, for illustration, we apply these results to investigate the asymptotic properties of several recently proposed Bayesian methods in general insurance.

Keywords

General insurance Nonparametric Bayes Posterior consistency Posterior robustness Property and casualty insurance 

Notes

Acknowledgements

The authors thank the Editor and two anonymous reviewers for their thoughtful comments and suggestions.

References

  1. 1.
    Barron A (1988) The exponential convergence of posterior probabilities with implications for Bayes estimators of density functions. Technical Report 7, Department of Statistics, University of Illinois, Champaign, ILGoogle Scholar
  2. 2.
    Barron A, Schervish MJ, Wasserman L (1999) The consistency of posterior distributions in nonparametric problems. Ann Stat 27:536–561MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berger JO (1985) Statistical decision theory and bayesian analysis, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  4. 4.
    Brockett PL, Chuang SL, Pitaktong U (2014) Generalized additive models and nonparametric regression. In: Predictive modeling applications in actuarial science. Cambridge University Press. pp 367–397Google Scholar
  5. 5.
    Bühlmann H (1967) Experience rating and credibility. ASTIN Bull 4:199–207CrossRefGoogle Scholar
  6. 6.
    Bühlmann H, Gisler A (2005) A course in credibility theory and its applications. Springer, New YorkMATHGoogle Scholar
  7. 7.
    Bühlmann H, Straub E (1970) Glaubwürdigkeit für Schadensätze. Mitteilungen der Vereinigung Schweizerischer Versicherungs-Mathematiker 70:111–133MATHGoogle Scholar
  8. 8.
    Bunke O, Milhaud X (1998) Asymptotic behavior of Bayes estimates under possibly incorrect models. Ann Stat 26(2):617–644MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cai X, Wen L, Wu X, Zhou X (2015) Credibility estimation of distribution functions with applications to experience rating in general insurance. N Am Actuar J 19(4):311–335MathSciNetCrossRefGoogle Scholar
  10. 10.
    Choi T, Ramamoorthi RV (2008) Remarks on consistency of posterior distributions. Pushing the limits of contemporary statistics: contributions in honor of Jayanta K. Ghosh. Inst Math Stat Collect 3:170–186CrossRefGoogle Scholar
  11. 11.
    Choi T, Schervish M (2007) On posterior consistency in nonparametric regression problems. J Multivar Anal 98:1969–1987MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    de Alba E (2002) Bayesian estimation of outstanding claim reserves. N Am Actuar J 6(4):1–20MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    de Alba E (2006) Claim reserving when there are negative values in the runoff triangle. N Am Actuar J 10(3):45–59MathSciNetCrossRefGoogle Scholar
  14. 14.
    De Blasi P, Walker SG (2013) Bayesian asymptotics with misspecified models. Stat Sin 23:169–187MathSciNetMATHGoogle Scholar
  15. 15.
    Diaconis P, Freedman D (1986) On the consistency of Bayes estimates. Ann Stat 14(1):1–26MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Doob JL (1949) Application of the theory of martingales. In: Le Calcul des Probabilités et ses applications. Colloques Internationaux du Centre National de la Recherche Scientifique. Paris. pp 23–27Google Scholar
  17. 17.
    Escoto B (2013) Bayesian claim severity with mixed distributions. Variance 7(2):110–122Google Scholar
  18. 18.
    Fellingham GW, Kottas A, Hartman BM (2015) Bayesian nonparametric predictive modeling of group health claims. Insur Math Econ 60:1–10MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ferguson TS (1973) Bayesian analysis of some nonparametric problems. Ann Stat 1:209–230MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Gangopadhyay A, Gau WC (2007) Bayesian nonparametric approach to credibility modeling. Ann Actuar Sci 2(I):91–114CrossRefGoogle Scholar
  21. 21.
    Ghosal S (2010) The Dirichlet process, related priors and posterior asymptotics. In: Nils Hjort, Chris Holmes, Peter Müller, and Stephen G. Walker (eds) Bayesian nonparametrics. Cambridge University Press, Cambridge. pp 35–79Google Scholar
  22. 22.
    Ghosal S, Ghosh JK, Ramamoorthi RV (1999) Posterior consistency of Dirichlet mixtures in density estimation. Ann Stat 27:143–158MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ghosh JK, Ramamoorthi RV (2003) Bayesian nonparametrics. Springer, New YorkMATHGoogle Scholar
  24. 24.
    Hong L, Martin R (2016) Discussion on “Credibility Estimation of Distribution Functions with Applications to Experience Rating in General Insurance”. N Am Actuar J 20(1):95–98MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hong L, Martin R (2017) A flexible Bayesian nonparametric model for predicting future insurance claims. N Am Actuar J. doi: 10.1080/10920277.2016.1247720
  26. 26.
    Jara A, Hanson T, Quintana F, Müller P, Rosner G (2011) DPpackage: Bayesian semi- and nonparametric modeling in R. J Stat Softw 40(1):1–30Google Scholar
  27. 27.
    Jeon Y, Kim JHT (2013) A gamma kernel density estimation for insurance loss data. Insur Math Econ 53:569–579MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kaas R, Dannenburg D, Goovaerts M (1997) Exact credibility for weighted observations. ASTIN Bull 27(2):287–295CrossRefGoogle Scholar
  29. 29.
    Kass RE, Wasserman L (1996) The selection of prior distributions by formal rules. J Am Stat Assoc 91:1343–1370CrossRefMATHGoogle Scholar
  30. 30.
    Kleijn BJK, van der Vaart AW (2006) Misspecification in infinite-dimensional Bayesian statistics. Ann Stat 34(2):837–877MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Klugman SA (1992) Bayesian statistics in actuarial science with emphasis on credibility. Kluwer, BostonCrossRefMATHGoogle Scholar
  32. 32.
    Klugman SA, Panjer HH, Willmot GE (2008) Loss models: from data to decisions, 3rd edn. Wiley, HobokenCrossRefMATHGoogle Scholar
  33. 33.
    Kuo H (1975) Gaussian measures in banach spaces. Springer, New YorkCrossRefMATHGoogle Scholar
  34. 34.
    Lau WJ, Siu TK, Yang H (2006) On Bayesian mixture credibility. ASTIN Bull 36(2):573–588MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lee SCK, Lin XS (2010) Modeling and evaluating insurance losses via mixtures of Erlang distributions. N Am Actuar J 14(1):107–130MathSciNetCrossRefGoogle Scholar
  36. 36.
    Lehmann EL (2006) Nonparametrics: statistical methods based on ranks, revised first edition. Springer, New YorkGoogle Scholar
  37. 37.
    Lehmann EL, Casella G (1998) Theory of point estimation, 2nd edn. New York, SpringerMATHGoogle Scholar
  38. 38.
    Makov UE, Smith AFM, Liu YH (1996) Bayesian methods in actuarial science. Statistician 45(4):503–515CrossRefGoogle Scholar
  39. 39.
    Makov UE (2001) Principal applications of Bayesian methods in actuarial science. N Am Actuar J 5(4):53–57MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Merz M, Wüthrich MV (2010) Paid-incurred chain claims reserving methods. Insur Math Econ 46:568–579MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Ntzoufras I, Dellaportas P (2002) Bayesian modeling of outstanding liabilities incorporating claim count uncertainty. N Am Actuar J 6(1):113–125MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Pan M, Wang R, Wu X (2008) On the consistency of credibility premiums regarding Esscher principles. Insur Math Econ 42:119–126MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Ramamoorthi RV, Sriram K, Martin R (2015) On posterior concentration in misspecified models. Bayesian Anal 10:759–789MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Rempala GA, Derrig RA (2005) Modeling hidden exposures in claim severity via the EM algorithm. N Am Actuar J 9(2):108–128MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Schervish MJ (1995) Theory of statistics. Springer, New YorkCrossRefMATHGoogle Scholar
  46. 46.
    Schmidt KD (1991) Convergence of Bayes and credibility premiums. ASTIN Bull 20(2):167–172CrossRefGoogle Scholar
  47. 47.
    Schwartz L (1965) On bayes procedures. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4:10–26MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Scollnik DPM (2001) Actuarial modeling with MCMC and BUGS. N Am Actuar J 5(2):96–124MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Shen X, Wasserman L (2001) Rates of convergence of posterior distributions. Ann Stat 29(3):687–714MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Shi P, Basu S, Meyers GG (2012) A Bayesian lognormal model for multivariate loss reserving. N Am Actuar J 16(1):1–29MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Shyamalkumar ND (1996) Cyclic \(I_0\) projections and its applications in statistics. Purdue University Technical Report \(\#\) 96–24Google Scholar
  52. 52.
    Tokdar ST (2006) Posterior consistency of Dirichlet location-scale mixture of normals in density estimation and regression. Sankhyā 67(4):90–110MathSciNetMATHGoogle Scholar
  53. 53.
    van der Geer S (2003) Asymptotic theory for maximum likelihood in nonparametric mixture models. Comput Stat Data Anal 41:453–464MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    van der Vaart AW, Wellner J (1996) Weak convergence and empirical processes: with applications to statistics. Springer, New YorkCrossRefMATHGoogle Scholar
  55. 55.
    Wald A (1949) Note on the consistency of the maximum likelihood estimate. Ann Math Stat 20:595–601MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Walker SG (2003) On sufficient conditions for Bayesian consistency. Biometrika 90:482–488MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Walker SG (2004) New approaches to Bayesian consistency. Ann Stat 32:2028–2043MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Werner G, Modlin C (2010) Basic ratemaking. Casualty Actuarial Society, ArlingtonGoogle Scholar
  59. 59.
    Wu Y, Ghosal S (2008) Kullback Leibler property of kernel mixture priors in Bayesian density estimation. Electron J Stat 2:298–331MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Wüthrich MV (2012) “A Bayesian log-normal model for multivariate loss reserving, Peng Shi, Sanjib Basu, and Glenn G. Meyers, March 2012”. N Am Actuar J 16(2):398–401MathSciNetCrossRefMATHGoogle Scholar
  61. 61.
    Zhang Y, Dukic V (2013) Predicting multivariate insurance loss payments under the Bayesian copula framework. J Risk Insur 80(4):891–919CrossRefGoogle Scholar

Copyright information

© EAJ Association 2017

Authors and Affiliations

  1. 1.Department of MathematicsRobert Morris UniversityMoonUSA
  2. 2.Department of StatisticsNorth Carolina State UniversityRaleighUSA

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