Skip to main content

Exchangeable mortality projection

Abstract

In this study we derive a novel multi-population Lee-Carter type model. Specifically, we extend the Bayesian model in Czado et al. (Insur Math Econ 36:260–284, 2005) to allow exchangeability between parameters of a group of m populations. In a validation-based examination, the proposed model is found to be beneficial for several examined countries. Also, we examine changes in forecasting ability due to varying calibration periods. Our results suggest that mortality rates from a distant past are inferior to those from a more recent past.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

    Achana FA, Cooper NJ, Bujkiewicz S, Hubbard SJ, Kendrick D, Jones DR, Sutton AJ (2014) Network meta-analysis of multiple outcome measures accounting for borrowing of information across outcomes. BMC Med Res Methodol 14(1):1

    Google Scholar 

  2. 2.

    Ahcan A, Medved D, Olivieri A, Pitacco E (2014) Forecasting mortality for small populations by mixing mortality data. Insur Math Econ 54:12–27

    MathSciNet  Google Scholar 

  3. 3.

    Alexander M, Zagheni E, Barbieri M (2017) A flexible Bayesian model for estimating subnational mortality. Demography 54(6):2025–2041

    Google Scholar 

  4. 4.

    Alkema L, Raftery AE, Gerland P, Clark SJ, Pelletier F, Buettner T, Heilig GK (2011) Probabilistic projections of the total fertility rate for all countries. Demography 48(3):815–839

    Google Scholar 

  5. 5.

    Antonio K, Bardoutsos A, Ouburg W (2015) Bayesian Poisson log-bilinear models for mortality projections with multiple populations. Eur Actuar J 5(2):245–281

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Biatat VD, Currie ID (2010) Joint models for classification and comparison of mortality in different countries. In: Proceedings of 25rd international workshop on statistical modelling, Glasgow, pp 89–94

  7. 7.

    Booth H, Maindonald J, Smith L (2002) Applying Lee–Carter under conditions of variable mortality decline. Popul Stud 56(3):325–336

    Google Scholar 

  8. 8.

    Box GEP, Tiao GC (1968) Bayesian estimation of means for the random effect model. J Am Stat Assoc 63:174–181

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Brouhns N, Denuit M, Vermunt JK et al (2002) Measuring the longevity risk in mortality projections. Bull Swiss Assoc Actuar 2:105–130

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Brouhns N, Denuit M, Vermunt JK (2002) A Poisson log-bilinear regression approach to the construction of projected lifetables. Insur Math Econ 31(3):373–393

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Butt Z, Haberman S (2009) ilc: a collection of R functions for fitting a class of Lee-Carter mortality models using iterative fitting algorithms. http://openaccess.city.ac.uk/2321/

  12. 12.

    Cairns AJG, Blake D, Dowd K, Coughlan GD, Khalaf-Allah M (2011) Bayesian stochastic mortality modelling for two populations. Astin Bull 41(01):29–59

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Carter LR, Lee RD (1992) Modeling and forecasting US sex differentials in mortality. Int J Forecast 8(3):393–411

    Google Scholar 

  14. 14.

    Carter LR, Prskawetz A (2001) Examining structural shifts in mortality using the Lee–Carter method. Methoden und Ziele 39

  15. 15.

    Coelho E, Nunes L (2013) Cohort effects and structural changes in the mortality trend. In: Technical report. Working paper. http://www.unece.org/fileadmin/DAM/stats/documents/ece/ces/ge.11/2013/WP5.1.pdf.

  16. 16.

    Czado C, Delwarde A, Denuit M (2005) Bayesian Poisson log-bilinear mortality projections. Insur Math Econ 36(3):260–284

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Danesi IL, Haberman S, Millossovich P (2015) Forecasting mortality in subpopulations using Lee–Carter type models: a comparison. Insur Math Econ 62:151–161

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Debón A, Montes F, Martínez-Ruiz F (2011) Statistical methods to compare mortality for a group with non-divergent populations: an application to Spanish regions. Eur Actuar J 1(2):291–308

    MathSciNet  Google Scholar 

  19. 19.

    De Finetti B (1937) La prévision: ses lois logiques, ses sources subjectives. Annales de l’institut Henri Poincaré 7:1–68

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Delwarde A, Denuit M, Guillén M, Vidiella-i Anguera A (2006) Application of the Poisson log-bilinear projection model to the G5 mortality experience. Belgian Actuar Bull 6(1):54–68

    MATH  Google Scholar 

  21. 21.

    Denuit M, Goderniaux A (2005) Closing and projecting lifetables using log-linear models. Bull Swiss Assoc Actuari, p 29

  22. 22.

    Fuse M, Yamasue E, Reck BK, Graedel TE (2011) Regional development or resource preservation? A perspective from Japanese appliance exports. Ecol Econ 70(4):788–797

    Google Scholar 

  23. 23.

    Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans Pattern Anal Mach Intelli 6:721–741

    MATH  Google Scholar 

  24. 24.

    Gilks WR, Wild P (1992) Adaptive rejection sampling for Gibbs sampling. J R Stat Soc 41(2):337–348

    MATH  Google Scholar 

  25. 25.

    Gill J (2007) Bayesian methods: a social and behavioral sciences approach. CRC press, Cambridge

    MATH  Google Scholar 

  26. 26.

    Gottardo R, Li W, Johnson WE, Liu XS (2008) A flexible and powerful Bayesian hierarchical model for ChIP-chip experiments. Biometrics 64(2):468–478

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Greco F, Scalone F (2013) A space-time extension of the Lee–Carter model in a hierarchical Bayesian framework: modelling and forecasting provincial mortality in Italy. https://pdfs.semanticscholar.org/62c2/d02a22557042def54fa1ce1467808835a05a.pdf

  28. 28.

    Halstead BJ, Wylie GD, Coates PS, Valcarcel P, Casazza ML (2012) Bayesian shared frailty models for regional inference about wildlife survival. Anim Conserv 15(2):117–124

    Google Scholar 

  29. 29.

    Hong H, Chu H, Zhang J, Carlin BP (2016) A Bayesian missing data framework for generalized multiple outcome mixed treatment comparisons. Res Synth Methods 7(1):6–22

    Google Scholar 

  30. 30.

    Human Mortality Database. University of California Berkeley (USA) and Max Planck Institute for Demographic Research (Germany). www.mortality.org. Accessed 20 June 2016

  31. 31.

    Janssen F, Kunst A (2007) The choice among past trends as a basis for the prediction of future trends in old-age mortality. Popul Stud 61(3):315–326

    Google Scholar 

  32. 32.

    Kasim RM, Raudenbush SW (1998) Application of Gibbs sampling to nested variance components models with heterogeneous within-group variance. J Educ Behav Stat 23(2):93–116

    Google Scholar 

  33. 33.

    Kleinow T (2015) A common age effect model for the mortality of multiple populations. Insur Math Econ 63:147–152

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Kogure A, Kurachi Y (2010) A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions. Insur Math Econ 46(1):162–172

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Kogure A, Kitsukawa K, Kurachi Y (2009) A Bayesian comparison of models for changing mortalities toward evaluating longevity risk in Japan. Asia Pac J Risk Insur 3(2)

  36. 36.

    Lee RD, Carter LR (1992) Modeling and forecasting US mortality. J Am Stat Assoc 87(419):659–671

    MATH  Google Scholar 

  37. 37.

    Lee R, Miller T (2001) Evaluating the performance of the Lee–Carter method for forecasting mortality. Demography 38(4):537–549

    MathSciNet  Google Scholar 

  38. 38.

    Leonard T (1972) Bayesian methods for binomial data. Biometrika 59(3):581–589

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Li J (2013) A Poisson common factor model for projecting mortality and life expectancy jointly for females and males. Popul Stud 67(1):111–126

    Google Scholar 

  40. 40.

    Li J et al (2014) An application of MCMC simulation in mortality projection for populations with limited data. Demogr Res 30:1–48

    Google Scholar 

  41. 41.

    Li JS-H, Hardy MR (2011) Measuring basis risk in longevity hedges. N Am Actuar J 15(2):177–200

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Li N, Lee R (2005) Coherent mortality forecasts for a group of populations: an extension of the Lee–Carter method. Demography 42(3):575–594

    Google Scholar 

  43. 43.

    Lindley DV (1972) Bayesian statistics: a review. SIAM, New Delhi

    MATH  Google Scholar 

  44. 44.

    Lindley DV, Novick MR (1981) The role of exchangeability in inference. Ann Stat 9:45–58

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Li N, Lee R, Tuljapurkar S (2004) Using the Lee–Carter method to forecast mortality for populations with limited data*. Int Stat Rev 72(1):19–36

    MATH  Google Scholar 

  46. 46.

    Li H, De Waegenaere A, Melenberg B (2015) The choice of sample size for mortality forecasting: a Bayesian learning approach. Insur Math Econ 63:153–168

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Maiti T (1998) Hierarchical Bayes estimation of mortality rates for disease mapping. J Stat Plan Inference 69(2):339–348

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Owen RK, Tincello DG, Keith RA (2015) Network meta-analysis: development of a three-level hierarchical modeling approach incorporating dose-related constraints. Value Health 18(1):116–126

    Google Scholar 

  49. 49.

    Papadatou E, Pradel R, Schaub M, Dolch D, Geiger H, Ibañez C, Kerth G, Popa-Lisseanu A, Schorcht W, Teubner J et al (2012) Comparing survival among species with imperfect detection using multilevel analysis of mark–recapture data: a case study on bats. Ecography 35(2):153–161

    Google Scholar 

  50. 50.

    Pedroza C (2006) A Bayesian forecasting model: predicting US male mortality. Biostatistics 7(4):530–550

    MATH  Google Scholar 

  51. 51.

    Renshaw AE, Haberman S (2003) Lee–Carter mortality forecasting with age-specific enhancement. Insur Math Econ 33(2):255–272

    MathSciNet  MATH  Google Scholar 

  52. 52.

    Russolillo M, Giordano G, Haberman S (2011) Extending the Lee–Carter model: a three-way decomposition. Scand Actuar J 2011(2):96–117

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Shair S, Purcal S, Parr N (2017) Evaluating extensions to coherent mortality forecasting models. Risks 5(1):16

    Google Scholar 

  54. 54.

    Shang HL, Hyndman RJ, Booth H, et al. (2010) A comparison of ten principal component methods for forecasting mortality rates

  55. 55.

    Smith AFM (1973) A general Bayesian linear model. J R Stat Soc Ser B Methodol 35:67–75

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Son YS, Oh M (2006) Bayesian analysis of the two-parameter gamma distribution. Commun Stat Simul Comput 35:285–293

    MATH  Google Scholar 

  57. 57.

    Tsionas EG (2001) Exact inference in four-parameter generalized gamma distributions. Commun Stat Theory Methods 30:747–756

    MathSciNet  MATH  Google Scholar 

  58. 58.

    Tuljapurkar S, Li N, Boe C (2000) A universal pattern of mortality decline in the G7 countries. Nature 405(6788):789–792

    Google Scholar 

  59. 59.

    Villegas AM, Haberman S (2014) On the modeling and forecasting of socioeconomic mortality differentials: an application to deprivation and mortality in England. N Am Actuar J 18(1):168–193

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Villegas AM, Haberman S, Kaishev VK, Millossovich P (2017) A comparative study of two-population models for the assessment of basis risk in longevity hedges. ASTIN Bull J IAA 47(3):631–679

    MathSciNet  MATH  Google Scholar 

  61. 61.

    Warren FC, Abrams KR, Sutton AJ (2014) Hierarchical network meta-analysis models to address sparsity of events and differing treatment classifications with regard to adverse outcomes. Stat Med 33(14):2449–2466

    MathSciNet  Google Scholar 

  62. 62.

    Wilmoth J, Valkonen T (2001) A parametric representation of mortality differentials over age and time. In: Fifth seminar of the EAPS working group on differentials in health, morbidity and mortality in Europe, Pontignano, Italy

  63. 63.

    Wiśniowski A, Smith PWF, Bijak J, Raymer J, Forster JJ (2015) Bayesian population forecasting: extending the Lee-Carter method. Demography 52(3):1035–1059

    Google Scholar 

  64. 64.

    Yang B, Li J, Balasooriya U (2016) Cohort extensions of the Poisson common factor model for modelling both genders jointly. Scand Actuar J 2016(2):93–112

    MathSciNet  MATH  Google Scholar 

  65. 65.

    Yang SS, Wang C-W (2013) Pricing and securitization of multi-country longevity risk with mortality dependence. Insur Math Econ 52(2):157–169

    MathSciNet  MATH  Google Scholar 

  66. 66.

    Yang S, Yue JC, Yeh Y-Y (2011) Coherent mortality modeling for a group of populations. In: Living to 100 symposium

  67. 67.

    Zhou R, Wang Y, Kaufhold K, Li JS-H, Tan KS (2014) Modeling period effects in multi-population mortality models: applications to Solvency II. N Am Actuar J 18(1):150–167

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vered Shapovalov.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Shapovalov, V., Landsman, Z. & Makov, U. Exchangeable mortality projection. Eur. Actuar. J. 11, 113–133 (2021). https://doi.org/10.1007/s13385-020-00255-w

Download citation

Keywords

  • Bayesian
  • Lee–Carter methodology
  • Mortality forecasting
  • Optimal calibration period
  • Multi-population model
  • Validation-based approach