A Neural Approach to Improve the Lee-Carter Mortality Density Forecasts

REFERENCES

• Abadi, M., A. Aggarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, et al. 2015. TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems. https://www.tensorflow.org/
   Google Scholar
• Aggarwal, C. C. 2018. Neural networks and deep learning. A textbook. Cham: Springer
   Google Scholar
• Booth, H., and L. Tickle. 2008. Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science 3 (1–2):3–43. doi:10.1017/S1748499500000440
   Google Scholar
• Breiman, L. 1996. Bagging predictors. Machine Learning 24:123–40. doi:10.1007/BF00058655
   Web of Science ®Google Scholar
• Brouhns, N., M. Denuit, and I. van Keilegom. 2005. Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scandinavian Actuarial Journal 3:212–24. doi:10.1080/03461230510009754
   Google Scholar
• Brouhns, N., M. Denuit, and J. Vermunt. 2002. A Poisson log-bilinear regression approach to the construction of projected life tables. Insurance: Mathematics and Economics 3:373–93.
   Google Scholar
• Cairns, A. J. G., D. Blake, and K. Dowd. 2006. A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance 73:687–718. doi:10.1111/j.1539-6975.2006.00195.x
   Web of Science ®Google Scholar
• Cairns, A. J. G., D. Blake, K. Dowd, G. D. Coughlan, D. Epstein, and M. Khalaf-Allah. 2011. Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics 48 (3):355–67. doi:10.1016/j.insmatheco.2010.12.005
   Web of Science ®Google Scholar
• Cairns, A. J. G., D. Blake, K. Dowd, G. D. Coughlan, D. Epstein, A. Ong, and I. Balevich. 2009. A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal 13:1–35. doi:10.1080/10920277.2009.10597538
   Google Scholar
• Camarda, C. G. 2019. Smooth constrained mortality forecasting. Demographic Research 41:1091–130. doi:10.4054/DemRes.2019.41.38
   Web of Science ®Google Scholar
• Carney, J. G., P. Cunningham, and U. Bhagwan. 1999. Confidence and prediction intervals for neural network ensembles. In Proceeding of the International Joint Conference on Neural Networks, 1215–18. Washington DC, U.S.: IEEE Computer Society Press.
   Google Scholar
• Chollet, F. 2017. R Interface to Keras, GitHub, https://github.com/rstudio/keras
   Google Scholar
• Currie, I. D. 2017. On fitting generalized linear and non-linear models of mortality. Scandinavian Actuarial Journal 4:356–83. doi:10.1080/03461238.2014.928230
   Google Scholar
• Currie, I. D., M. Durban, and P. H. C. Eilers. 2004. Smoothing and forecasting mortality rates. Statistical Modelling 4:279–98. doi:10.1191/1471082X04st080oa
   Web of Science ®Google Scholar
• D’Amato, V., E. Di Lorenzo, S. Haberman, M. Russolillo, and M. Sibillo. 2011. The Poisson log-bilinear Lee-Carter model. North American Actuarial Journal 15 (2):315–33. doi:10.1080/10920277.2011.10597623
   Google Scholar
• D’Amato, V., S. Haberman, G. Piscopo, and M. Russolillo. 2012. Modelling dependent data for longevity projections. Insurance: Mathematics and Economics 51 (3):694–701. doi:10.1016/j.insmatheco.2012.09.008
   Web of Science ®Google Scholar
• D’Amato, V., S. Haberman, and M. Russolillo. 2012. The stratified sampling bootstrap for measuring the uncertainty in mortality forecasts. Methodology Computing in Applied Probability 14:135–48. doi:10.1007/s11009-011-9225-z
   Web of Science ®Google Scholar
• Deprez, P., P. V. Shevchenko, and M. V. Wüthrich. 2017. Machine learning techniques for mortality modeling. European Actuarial Journal 7:337–52. doi:10.1007/s13385-017-0152-4
   Web of Science ®Google Scholar
• Dowd, K., D. Blake, A. J. G. Cairns, G. D. Coughlan, D. Epstein, and M. Khalaf-Allah. 2010. Backtesting stochastic mortality models: An ex-post valuation of multi-period-ahead density forecasts. North American Actuarial Journal 14:281–98. doi:10.1080/10920277.2010.10597592
   Google Scholar
• Efron, B., and R. Tibshirani. 1993. An introduction to the bootstrap. New York: Chapman & Hall.
   Google Scholar
• Geman, S., E. Bienenstock, and R. Doursat. 1992. Neural networks and the bias/variance dilemma. Neural Computation 4 (1):1–58. doi:10.1162/neco.1992.4.1.1
   Web of Science ®Google Scholar
• Goodfellow, I., Y. Bengio, and A. Courville. 2016. Deep learning. Cambridge, MA, U. S.: MIT Press.
   Google Scholar
• Hainaut, D. 2018. A neural-network analyzer for mortality forecast. ASTIN Bulletin 48 (2):481–508. doi:10.1017/asb.2017.45
   Web of Science ®Google Scholar
• Heskes, T. 1997. Practical confidence and prediction intervals. In Advances in neural information processing systems, vol. 9, 176–82. Cambridge, MA, U. S.: MIT Press.
   Google Scholar
• Hocreiter, S., and J. Schmidhuber. 1997. Long short-term memory. Neural Computation 9 (8):1735–80. doi:10.1162/neco.1997.9.8.1735
   PubMed Web of Science ®Google Scholar
• Human Mortality Database. 2018. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). https://www.mortality.org.
   Google Scholar
• Hunt, A., and D. Blake. 2014. A general procedure for constructing mortality models. North American Actuarial Journal 18 (1): 116–38. doi:10.1080/10920277.2013.852963
   Google Scholar
• Hyndman, R. J., and Y. Khandakar. 2008. Automatic time series forecasting: The forecast package for R. Journal of Statistical Software 26 (3):1–22. doi:10.18637/jss.v027.i03
   PubMed Web of Science ®Google Scholar
• Kasiviswanathan, K. S., and K. P. Sudheer. 2013. Quantification of the predictive uncertainty of artificial neural network based river flow forecast models. Stochastic Environmental Research and Risk Assessment 27:137–46. doi:10.1007/s00477-012-0600-2
   Web of Science ®Google Scholar
• Khosravi, A., S. Nahavandi, D. Creighton, and A. F. Atiya. 2011. Comprehensive review of neural network–based prediction intervals and new advances. IEEE Transactions on Neural Networks 22 (9):1341–56. doi:10.1109/TNN.2011.2162110
   PubMedGoogle Scholar
• Khosravi, A., S. Nahavandi, D. Srinivasan, and R. Khosravi. 2015. Constructing optimal prediction intervals by using neural networks and bootstrap method. IEEE Transactions on Neural Networks and Learning Systems 26 (8):1810–15. doi:10.1109/TNNLS.2014.2354418
   PubMed Web of Science ®Google Scholar
• Koissi, M., A. Shapiro, and G. Hognas. 2006. Evaluating and extending the Lee-Carter model for mortality forecasting confidence interval. Insurance: Mathematics and Economics 1:1–20.
   Google Scholar
• Lee, R. D., and L. R. Carter. 1992. Modeling and forecasting U.S. mortality. Journal of the American Statistical Association 87 (419):659–71. doi:10.1080/01621459.1992.10475265
   Web of Science ®Google Scholar
• Levantesi, S., and A. Nigri. 2020. A random forest algorithm to improve the Lee-Carter mortality forecasting: Impact on q-forward. Soft Computing 24:8553–67. doi:10.1007/s00500-019-04427-z
   Web of Science ®Google Scholar
• Levantesi, S., A. Nigri, and G. Piscopo. 2021. Clustering-based simultaneous forecasting of life expectancy time series through long-short term memory neural networks. International Journal of Approximate Reasoning 140:282–97. doi:10.1016/j.ijar.2021.10.008
   Web of Science ®Google Scholar
• Levantesi, S., and V. Pizzorusso. 2019. Application of machine learning to mortality modeling and forecasting. Risks 7 (1):26.
   Web of Science ®Google Scholar
• Li, J., M. Hardy, and K. Tan. 2009. Uncertainty in mortality forecasting: An extension to the classical Lee-Carter approach. ASTIN Bulletin 39 (1):137–64. doi:10.2143/AST.39.1.2038060
   Web of Science ®Google Scholar
• Li, K., R. Wang, H. Lei, T. Zhang, Y. Liu, and X. Zheng. 2018. Interval prediction of solar power using an improved bootstrap method. Solar Energy 159:97–112. doi:10.1016/j.solener.2017.10.051
   Web of Science ®Google Scholar
• MacKay, D. J. C. 1992. A practical Bayesian framework for backpropagation networks. Neural Computation 4 (3):448–72. doi:10.1162/neco.1992.4.3.448
   Web of Science ®Google Scholar
• Makridakis, S., E. Spiliotis, and V. Assimakopoulos. 2020. The M4 Competition: 100,000 Time series and 61 forecasting methods. International Journal of Forecasting 36 (1):54–74. doi:10.1016/j.ijforecast.2019.04.014
   Web of Science ®Google Scholar
• Mazloumi, E., G. Rose, G. Currie, and S. Moridpour. 2011. Prediction intervals to account for uncertainties in neural network predictions: Methodology and application in bus travel time prediction. Engineering Applications of Artificial Intelligence 24 (3):534–42. doi:10.1016/j.engappai.2010.11.004
   Web of Science ®Google Scholar
• Mitchell, D., P. Brockett, R. Mendoza-Arriaga, and K. Muthuraman. 2013. Modeling and forecasting mortality rates. Insurance: Mathematics and Economics 52 (2):275–85. doi:10.1016/j.insmatheco.2013.01.002
   Web of Science ®Google Scholar
• Nair, V., and G. Hinton. 2010. Rectified linear units improve restricted Boltzmann machines. In Proceedings of the 27th International Conference on International Conference on Machine Learning, 807–14. Madison, WI: Omnipress.
   Google Scholar
• Nigri, A., E. Barbi, and S. Levantesi. 2021. The relationship between longevity and lifespan variation. Statistical Methods and Applications. doi:10.1007/s10260-021-00584-4
   Google Scholar
• Nigri, A., S. Levantesi, and M. Marino. 2020. Life expectancy and lifespan disparity forecasting: A long short-term memory approach. Scandinavian Actuarial Journal 2:110–33.
   Web of Science ®Google Scholar
• Nigri, A., S. Levantesi, M. Marino, S. Scognamiglio, and F. Perla. 2019. A deep learning integrated Lee-Carter model. Risks 7 (1):33.
   Web of Science ®Google Scholar
• Nix, D. A., and A. S. Weigend. 1994. Estimating the mean and variance of the target probability distribution. In Proceeding of IEEE International Conference on Neural Networks, vol. 1, 55–60. Washington DC, U. S.: IEEE Computer Society Press.
   Google Scholar
• Oeppen, J., and J. W. Vaupel. 2006. The linear rise in the number of our days. In Perspectives on mortality forecasting. III: The linear rise in life expectancy: history and prospects, vol. 3, ed. T. Bengtsson, 9–18. Stockholm: Swedish Social Insurance Agency.
   Google Scholar
• Pascanu, R., T. Mikolov, and Y. Bengio. 2013. On the difficulty of training recurrent neural networks. In Proceedings of Machine Learning Research 28 (3): vol. 28, 1310–18.
   Google Scholar
• Perla, F., R. Richman, S. Scognamiglio, and M. Wüthrich. 2021. Time-series forecasting of mortality rates using deep learning. Scandinavian Actuarial Journal 7:572–98.
   Web of Science ®Google Scholar
• Plat, R. 2009. On stochastic mortality modeling. Insurance: Mathematics and Economics 45:393–404. doi:10.1016/j.insmatheco.2009.08.006
   Web of Science ®Google Scholar
• R Core Team. 2020. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing.
   Google Scholar
• Rau, R., E. Soroko, D. Jasilionis, and J. W. Vaupel. 2008. Continued reductions in mortality at advanced ages. Population and Development Review 34:747–68. doi:10.1111/j.1728-4457.2008.00249.x
   Web of Science ®Google Scholar
• Renshaw, A. E., and S. Haberman. 2006. A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics 38 (3):556–70. doi:10.1016/j.insmatheco.2005.12.001
   Web of Science ®Google Scholar
• Richman, R., and M. V. Wüthrich.. 2021. A neural network extension of the Lee-Carter model to multiple populations. Annals of Actuarial Science 15 (2):346–66.
   Google Scholar
• Rumelhart, D., G. Hinton, and R. Williams. 1986. Learning representations by back-propagating errors. Nature 323:533–36. doi:10.1038/323533a0
   Web of Science ®Google Scholar
• Schäfer, A. M., and H. G. Zimmermann. 2007. Recurrent neural networks are universal approximators. International Journal of Neural Systems 17 (4):253–63. doi:10.1142/S0129065707001111
   PubMed Web of Science ®Google Scholar
• Schnürch, S., and R. Korn. 2021. Point and interval forecasts of death rates using neural networks. ASTIN Bulletin 8 (52):333–60.
   Web of Science ®Google Scholar
• Tibshirani, R. 1996. A comparison of some error estimates for neural network models. Neural Computation 8:152–63. doi:10.1162/neco.1996.8.1.152
   Web of Science ®Google Scholar
• Villegas, A. M., V. K. Kaishev, and P. Millossovich. 2018. StMoMo: An R Package for Stochastic Mortality Modeling. Journal of Statistical Software 84 (3):1–38. doi:10.18637/jss.v084.i03
   Google Scholar
• Wild, C. J., and G. A. F. Seber. 1989. Nonlinear regression. New York: Wiley.
   Google Scholar
• Zhou, Z. H., J. Wu, and W. Tang. 2002. Ensembling neural networks: Many could be better than all. Artificial Intelligence 137 (1–2):239–63. doi:10.1016/S0004-3702(02)00190-X
   Web of Science ®Google Scholar