## Wavelet-based feature extraction for mortality projection.(English)Zbl 1454.91190

Summary: Wavelet theory is known to be a powerful tool for compressing and processing time series or images. It consists in projecting a signal on an orthonormal basis of functions that are chosen in order to provide a sparse representation of the data. The first part of this article focuses on smoothing mortality curves by wavelets shrinkage. A chi-square test and a penalized likelihood approach are applied to determine the optimal degree of smoothing. The second part of this article is devoted to mortality forecasting. Wavelet coefficients exhibit clear trends for the Belgian population from 1965 to 2015, they are easy to forecast resulting in predicted future mortality rates. The wavelet-based approach is then compared with some popular actuarial models of Lee-Carter type estimated fitted to Belgian, UK, and US populations. The wavelet model outperforms all of them.

### MSC:

 91G05 Actuarial mathematics 91D20 Mathematical geography and demography 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems

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 [1] Besbeas, P., De Feis, I. and Sapatinas, T. (2004) A comparative simulation study of wavelet shrinkage estimators for Poisson counts. International Statistical Review, 72, 209-237. · Zbl 1211.62055 [2] Brouhns, N., Denuit, M. and Vermunt, J.K. (2002) A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31, 373-393. · Zbl 1074.62524 [3] Cairns, A.J.G., 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. [4] Cairns, A.J.G., 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. [5] Cochran, W.G. (1952) The $$\chi^2$$ test of goodness of fit. Annals of Mathematical Statistics, 23, 315-345. · Zbl 0047.13105 [6] Denuit, M., Hainaut, D. and Trufin, J. (2019a) Effective Statistical Learning Methods for Actuaries - Volume 1: GLM and Extensions. Springer Actuarial Lecture Notes Series. Springer Nature Switzerland. · Zbl 1426.62003 [7] Denuit, M., Hainaut, D. and Trufin, J. (2019b) Effective Statistical Learning Methods for Actuaries - Volume 3: Neural Networks and Extensions. Springer Actuarial Lecture Notes Series. Springer Nature Switzerland. and Legrand, C. (2018) Risk classification in life and health insurance: Extension to continuous covariates. European Actuarial Journal8, 245-255. · Zbl 1416.91170 [8] Donoho, D.L. and Johnstone, I.M. (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika81, 425-455. · Zbl 0815.62019 [9] Donoho, D.L. and Johnstone, I.M. (1995) Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Assocation90, 1200-1224. · Zbl 0869.62024 [10] Gbari, S., Poulain, M., Dal, L. and Denuit, M. (2017) Extreme value analysis of mortality at the oldest ages: A case study based on individual ages at death. North American Actuarial Journal, 21(3), 397-416. · Zbl 1414.91190 [11] Hastie, T., Tibshirani, R. and Friedman, J. (2016) The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Series in Statistics. Springer-Verlag New York. · Zbl 0973.62007 [12] Hyndman, R.J. and Ullah, Md.S. (2007) Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics and Data Analysis51, 4942-4956. · Zbl 1162.62434 [13] Jurado, F.M. and Sampere, I.B. (2019) Using wavelet techniques to approximate the subjacent risk of death. In: Modern Mathematics and Mechanics (eds. Sadovnichiy, V.A. and Zgurovsky, M.Z., Chapter 28, pp. 541-557. Springer International Publishing. · Zbl 1418.91251 [14] Lee, R.D. and Carter, L. (1992) Modelling and forecasting the time series of US mortality. Journal of the American Statistical Association87, 659-671. [15] Mallat, S.G. (1989a) Multiresolution approximations and wavelet orthonormal bases of $$L_2(\mathbb{R})$$ . Transactions of the American Mathematical Society315, 69-87. · Zbl 0686.42018 [16] Mallat, S.G. (1989b) A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence11, 674-693. · Zbl 0709.94650 [17] Morillas, F., Baeza, I. and Pavia, J.M. (2016) Risk of death: A two-step method using wavelets and piecewise harmonic interpolation. Estadistica Espanola58, 245-264. [18] Nickolas, P. (2017) Wavelets: A Student Guide. Cambridge University Press. · Zbl 1372.42036 [19] Pitacco, E., Denuit, M., Haberman, S. and Olivieri, A. (2009) Modelling Longevity Dynamics for Pensions and Annuity Business. New York: Oxford University Press. · Zbl 1163.91005 [20] Renshaw, A.E. and Haberman, S. (2003) Lee-Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics33, 255-272. · Zbl 1103.91371 [21] Renshaw, A.E. and Haberman, S. (2006) A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics38, 556-570. · Zbl 1168.91418 [22] Renshaw, A.E., Haberman, S. and Hatzoupoulos, P. (1996) The modelling of recent mortality trends in United Kingdom male assured lives. British Actuarial Journal2, 449-477. [23] Strang, G. and Nguyen, T. (1996) Wavelets and Filter Banks. Wellesley, MA. · Zbl 1254.94002 [24] Tibshirani, R. (1996) Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society - Series B58, 267-288. · Zbl 0850.62538 [25] Wilmoth, J.R., Andreev, K., Jdanov, D., Glei, D.A. and Rie, T. (2019) Methods Protocol for the Human Mortality Database. https://www.mortality.org/Public/Docs/MethodsProtocol.pdf
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