## Neighbouring prediction for mortality.(English)Zbl 1480.91248

Summary: We propose a new neighbouring prediction model for mortality forecasting. For each mortality rate at age $$x$$ in year $$t$$, $$m_{x,t}$$, we construct an image of neighbourhood mortality data around $$m_{x,t}$$, that is, $$\mathcal{E}_{m_{x,t}}(x_1, x_2, s)$$, which includes mortality information for ages in $$[x-x_1, x+x_2]$$, lagging $$k$$ years $$(1 \leq k \leq s)$$. Combined with the deep learning model – convolutional neural network, this framework is able to capture the intricate nonlinear structure in the mortality data: the neighbourhood effect, which can go beyond the directions of period, age, and cohort as in classic mortality models. By performing an extensive empirical analysis on all the 41 countries and regions in the Human Mortality Database, we find that the proposed models achieve superior forecasting performance. This framework can be further enhanced to capture the patterns and interactions between multiple populations.

### MSC:

 91G05 Actuarial mathematics 68T07 Artificial neural networks and deep learning
Full Text:

### References:

 [1] Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., et al. (2016) Tensorflow: A system for large-scale machine learning. 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI’16), pp. 265-283. [2] Blake, D., Cairns, A., Coughlan, G., Dowd, K. and Macminn, R. (2013) The new life market. Journal of Risk and Insurance, 80(3), 501-558. [3] Blake, D., Macminn, R., Tsai, J.C. and Wang, J. (2018) Longevity risk and capital markets: The 2017-18 update. Pension Institute Discussion Paper PI-1908. · Zbl 07341011 [4] Bottou, L. and Bousquet, O. (2008) The tradeoffs of large scale learning. Advances in Neural Information Processing Systems (eds. M. Jordan, Y. LeCun and S. Solla), pp. 161-168. [5] Cairns, A.J., Blake, D. and Dowd, K. (2006) A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73(4), 687-718. [6] Cairns, A.J., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A. and Balevich, I. (2009) A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal, 13(1), 1-35. [7] Chen, H., Macminn, R. and Sun, T. (2015) Multi-population mortality model: A factor copula approach. Insurance: Mathematics and Economics, 63, 135-146. · Zbl 1348.91131 [8] Chollet, F.et al. (2018) Keras: The Python deep learning library. Astrophysics Source Code Library. [9] Dietterich, T.G. (2000) Ensemble methods in machine learning. International Workshop on Multiple Classifier Systems, pp. 1-15. Springer. [10] Dong, Y., Huang, F., Yu, H. and Haberman, S. (forthcoming) Multi-population mortality forecasting using tensor decomposition. Scandinavian Actuarial Journal, pp. 334-356. · Zbl 1454.91179 [11] Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D. and Khalaf-Allah, M. (2011) A gravity model of mortality rates for two related populations. North American Actuarial Journal, 15(2), 334-356. · Zbl 1228.91032 [12] Hainaut, D. (2018) A neural-network analyzer for mortality forecast. ASTIN Bulletin: The Journal of the IAA, 48(2), 481-508. · Zbl 1390.91186 [13] Hastie, T., Tibshirani, R., Friedman, J. and Franklin, J. (2005) The elements of statistical learning: Data mining, inference and prediction. The Mathematical Intelligencer, 27(2), 83-85. [14] Jarner, S.F. and Kryger, E.M. (2011) Modelling adult mortality in small populations: The SAINT model. ASTIN Bulletin, 41(2), 377-418. · Zbl 1239.91128 [15] Krizhevsky, A., Sutskever, I. and Hinton, G.E. (2012) Imagenet classification with deep convolutional neural networks. Advances in Neural Information Processing Systems, pp. 1097-1105. [16] Lecun, Y., Bengio, Y. and Hinton, G. (2015) Deep learning. Nature, 521(7553), 436-444. [17] Lecun, Y., Bottou, L., Bengio, Y. and Haffner, P. (1998) Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11), 2278-2324. [18] Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting US mortality. Journal of the American Statistical Association, 87(419), 659-671. · Zbl 1351.62186 [19] Li, J.S.-H., Chan, W.-S. and Zhou, R. (2017) Semicoherent multipopulation mortality modeling: The impact on longevity risk securitization. Journal of Risk and Insurance, 84(3), 1025-1065. [20] Li, N. and Lee, R. (2005) Coherent mortality forecasts for a group of population: An extension to the classical Lee-Carter approach. Demography, 42(3), 575-594. [21] Perla, F., Richman, R., Scognamiglio, S. and Wuthrich, M.V. (2021) Time-series forecasting of mortality rates using deep learning. Scandinavian Actuarial Journal, pp. 1-27. · Zbl 1471.91480 [22] Renshaw, A.E. and Haberman, S. (2006) A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38(3), 556-570. · Zbl 1168.91418 [23] Richman, R. (2018) AI in actuarial science. Available at SSRN 3218082. [24] Richman, R. and Wüthrich, M.V. (forthcoming) A neural network extension of the Lee-Carter model to multiple populations. Annals of Actuarial Science, pp. 1-21. [25] Wang, C.-W., Yang, S.S. and Huang, H.-C. (2015) Modeling multi-country mortality dependence and its application in pricing survivor index swaps—a dynamic copula approach. Insurance: Mathematics and Economics, 63, 30-39. · Zbl 1348.62249 [26] Wang, H.-C., Yue, C.-S.J. and Chong, C.-T. (2018) Mortality models and longevity risk for small populations. Insurance: Mathematics and Economics, 78, 351-359. · Zbl 1400.91254 [27] Zhou, R., Li, J.S.-H. and Tan, K.S. (2013) Pricing standardized mortality securitizations: A two-population model with transitory jump effects. Journal of Risk and Insurance, 80(3), 733-774. [28] Zhou, Y.-T. and Chellappa, R. (1988) Computation of optical flow using a neural network. IEEE International Conference on Neural Networks, vol. 1998, pp. 71-78. [29] Zhu, W., Tan, K.S. and Wang, C.-W. (2017) Modeling multicountry longevity risk with mortality dependence: A Lévy subordinated hierarchical Archimedean copulas approach. Journal of Risk and Insurance, 84(S1), 477-493.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.