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Smooth projection of mortality improvement rates: a Bayesian two-dimensional spline approach

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Abstract

This paper proposes a spline mortality model for generating smooth projections of mortality improvement rates. In particular, we follow the two-dimensional cubic B-spline approach developed by Currie et al. (Stat Model 4(4):279–298, 2004), and adopt the Bayesian estimation and LASSO penalty to overcome the limitations of spline models in forecasting mortality rates. The resulting Bayesian spline model not only provides measures of stochastic and parameter uncertainties, but also allows external opinions on future mortality to be consistently incorporated. The mortality improvement rates projected by the proposed model are smoothly transitioned from the historical values with short-term trends shown in recent observations to the long-term terminal rates suggested by external opinions. Our technical work is complemented by numerical illustrations that use real mortality data and external rates to showcase the features of the proposed model.

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References

  1. Billari FC, Graziani R, Melilli E (2012) Stochastic population forecasts based on conditional expert opinions. J R Stat Soc Ser A (Stat Soc) 175(2):491–511

    Article  MathSciNet  Google Scholar 

  2. Billari FC, Graziani R, Melilli E (2014) Stochastic population forecasting based on combinations of expert evaluations within the Bayesian paradigm. Demography 51(5):1933–1954

    Article  Google Scholar 

  3. Cairns AJ, Blake D, Dowd K (2006) A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. J Risk Insur 73(4):687–718

    Article  Google Scholar 

  4. Camarda CG (2019) Smooth constrained mortality forecasting. Demogr Res 41:1091–1130

    Article  Google Scholar 

  5. Camarda CG et al (2012) Mortalitysmooth: an R package for smoothing Poisson counts with p-splines. J Stat Softw 50(1):1–24

    Article  MathSciNet  Google Scholar 

  6. Chang L, Shi Y (2020) Dynamic modelling and coherent forecasting of mortality rates: a time-varying coefficient spatial-temporal autoregressive approach. Scand Actuar J 2020(9):843–863

    Article  MathSciNet  MATH  Google Scholar 

  7. Currie ID (2013) Smoothing constrained generalized linear models with an application to the Lee–Carter model. Stat Model 13(1):69–93

    Article  MathSciNet  MATH  Google Scholar 

  8. Currie ID, Durban M, Eilers PH (2004) Smoothing and forecasting mortality rates. Stat Model 4(4):279–298

    Article  MathSciNet  MATH  Google Scholar 

  9. Currie ID, Durban M, Eilers PH (2006) Generalized linear array models with applications to multidimensional smoothing. J R Stat Soc Ser B (Stat Methodol) 68(2):259–280

    Article  MathSciNet  MATH  Google Scholar 

  10. Delwarde A, Denuit M, Eilers P (2007) Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting: a penalized log-likelihood approach. Stat Model 7(1):29–48

    Article  MathSciNet  MATH  Google Scholar 

  11. DiMatteo I, Genovese CR, Kass RE (2001) Bayesian curve-fitting with free-knot splines. Biometrika 88(4):1055–1071

    Article  MathSciNet  MATH  Google Scholar 

  12. Dodd E, Forster JJ, Bijak J, Smith PW (2018) Smoothing mortality data: the English life tables, 2010–2012. J R Stat Soc Ser A (Stat Soc) 181(3):717–735

    Article  MathSciNet  Google Scholar 

  13. Dodd E, Forster JJ, Bijak J, Smith PW (2021) Stochastic modelling and projection of mortality improvements using a hybrid parametric/semi-parametric age-period-cohort model. Scand Actuar J 2:134–155

    Article  MathSciNet  MATH  Google Scholar 

  14. Dowd K, Cairns A, Blake D (2020) CBDX: a workhorse mortality model from the Cairns–Blake–Dowd family. Ann Actuar Sci 14(2):445–460

    Article  Google Scholar 

  15. Eilers PH, Marx BD (1996) Flexible smoothing with B-splines and penalties. Stat Sci 11(2):89–121

    Article  MathSciNet  MATH  Google Scholar 

  16. Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7(4):457–472

    Article  MATH  Google Scholar 

  17. Geweke JF (1991) Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, vol 148. Federal Reserve Bank of Minneapolis, Research Department, Minneapolis

    Google Scholar 

  18. Guibert Q, Lopez O, Piette P (2019) Forecasting mortality rate improvements with a high-dimensional VAR. Insur Math Econ 88:255–272

    Article  MathSciNet  MATH  Google Scholar 

  19. Haberman S, Renshaw A (2012) Parametric mortality improvement rate modelling and projecting. Insur Math Econ 50(3):309–333

    Article  MathSciNet  MATH  Google Scholar 

  20. Haberman S, Renshaw A (2013) Modelling and projecting mortality improvement rates using a cohort perspective. Insur Math Econ 53(1):150–168

    Article  MathSciNet  MATH  Google Scholar 

  21. Hilton J, Dodd E, Forster JJ, Smith PWF (2019) Projecting UK mortality by using Bayesian generalized additive models. J R Stat Soc Ser C (Appl Stat) 68(1):29–49

    Article  MathSciNet  Google Scholar 

  22. Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1):55–67

    Article  MATH  Google Scholar 

  23. Huang F, Browne B (2017) Mortality forecasting using a modified continuous mortality investigation mortality projections model for China I: methodology and country-level results. Ann Actuar Sci 11(1):20–45

    Article  Google Scholar 

  24. Hunt A, Villegas AM (2017) Mortality improvement rates: modeling, parameter uncertainty and robustness. Presented at the living to 100 symposium

  25. Hyndman RJ, Ullah MS (2007) Robust forecasting of mortality and fertility rates: a functional data approach. Comput Stat Data Anal 51(10):4942–4956

    Article  MathSciNet  MATH  Google Scholar 

  26. Lang S, Brezger A (2004) Bayesian p-splines. J Comput Graph Stat 13(1):183–212

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  28. Li H, Lu Y (2017) Coherent forecasting of mortality rates: a sparse vector-autoregression approach. ASTIN Bull J IAA 47(2):563–600

    Article  MathSciNet  MATH  Google Scholar 

  29. Li H, Shi Y (2021) Mortality forecasting with an age-coherent sparse var model. Risks 9(2):35

    Article  Google Scholar 

  30. Li JS-H, Hardy M, Tan KS (2010) Developing mortality improvement formulas: the Canadian insured lives case study. North Am Actuar J 14(4):381–399

    Article  MathSciNet  Google Scholar 

  31. Li JS-H, Liu Y (2020) The heat wave model for constructing two-dimensional mortality improvement scales with measures of uncertainty. Insur Math Econ 93:1–26

    Article  MathSciNet  MATH  Google Scholar 

  32. Li JS-H, Liu Y (2021) Recent declines in life expectancy: implication on longevity risk hedging. Insur Math Econ 99:376–394

    Article  MathSciNet  MATH  Google Scholar 

  33. Li JS-H, Zhou KQ, Zhu X, Chan W-S, Chan FW-H (2019) A Bayesian approach to developing a stochastic mortality model for China. J R Stat Soc Ser A (Stat Soc) 182(4):1523–1560

    Article  MathSciNet  Google Scholar 

  34. Luoma A, Puustelli A, Koskinen L (2012) A Bayesian smoothing spline method for mortality modelling. Ann Actuar Sci 6(2):284–306

    Article  Google Scholar 

  35. Park T, Casella G (2008) The Bayesian Lasso. J Am Stat Assoc 103(482):681–686

    Article  MathSciNet  MATH  Google Scholar 

  36. Pitt D, Li J, Lim TK (2018) Smoothing Poisson common factor model for projecting mortality jointly for both sexes. ASTIN Bull J IAA 48(2):509–541

    Article  MathSciNet  MATH  Google Scholar 

  37. Rabbi AMF, Mazzuco S (2021) Mortality forecasting with the Lee-Carter method: adjusting for smoothing and lifespan disparity. Eur J Popul 37(1):97–120

    Article  Google Scholar 

  38. Renshaw A, Haberman S (2021) Modelling and forecasting mortality improvement rates with random effects. Eur Actuar J 11:381–412

    Article  MathSciNet  MATH  Google Scholar 

  39. Renshaw AE, Haberman S (2003) On the forecasting of mortality reduction factors. Insur Math Econ 32(3):379–401

    Article  MATH  Google Scholar 

  40. Richards SJ (2020) A Hermite-spline model of post-retirement mortality. Scand Actuar J 2:110–127

    Article  MathSciNet  MATH  Google Scholar 

  41. Shang HL (2019) Dynamic principal component regression: application to age-specific mortality forecasting. ASTIN Bull J IAA 49(3):619–645

    Article  MathSciNet  MATH  Google Scholar 

  42. Speckman PL, Sun D (2003) Fully Bayesian spline smoothing and intrinsic autoregressive priors. Biometrika 90(2):289–302

    Article  MathSciNet  MATH  Google Scholar 

  43. Tang KH, Dodd E, Forster JJ (2021) Joint modelling of male and female mortality rates using adaptive p-splines. Ann Actuar Sci 16:119–135

    Article  Google Scholar 

  44. Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc Ser B (Methodol) 58(1):267–288

    MathSciNet  MATH  Google Scholar 

  45. Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc Ser B (Stat Methodol) 67(2):301–320

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the two anonymous reviewers for their constructive comments and suggestions that improved the quality of this paper. The authors also greatly appreciate the comments received from Hua Chen and other participants at the 24th International Congress on Insurance: Mathematics and Economics.

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Authors and Affiliations

  1. Southwestern University of Finance and Economics, Chengdu, China

    Xiaobai Zhu

  2. Arizona State University, Tempe, USA

    Kenneth Q. Zhou

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  1. Xiaobai Zhu
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Correspondence to Kenneth Q. Zhou.

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Zhu, X., Zhou, K.Q. Smooth projection of mortality improvement rates: a Bayesian two-dimensional spline approach. Eur. Actuar. J. 13, 277–305 (2023). https://doi.org/10.1007/s13385-022-00323-3

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  • DOI: https://doi.org/10.1007/s13385-022-00323-3

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