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WAVELET-BASED FEATURE EXTRACTION FOR MORTALITY PROJECTION

Published online by Cambridge University Press:  25 June 2020

Donatien Hainaut Open the ORCID record for Donatien Hainaut [Opens in a new window]  and
Michel Denuit
Donatien Hainaut*
Affiliation:
Institute of Statistics, Biostatistis and Actuarial Science - ISBA Louvain Institute of Data Analysis and Modeling - LIDAM UCLouvain B-1348 Louvain-la-Neuve, Belgium E-Mails: donatien.hainaut@uclouvain.be; michel.denuit@uclouvain.be
Michel Denuit
Affiliation:
Institute of Statistics, Biostatistis and Actuarial Science - ISBA Louvain Institute of Data Analysis and Modeling - LIDAM UCLouvain B-1348 Louvain-la-Neuve, Belgium E-Mails: donatien.hainaut@uclouvain.be; michel.denuit@uclouvain.be
*
E-Mail: donatien.hainaut@uclouvain.be
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Abstract

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.

Keywords

Discrete wavelet transformmortality smoothingPoisson regressionregularizationLassoG22C01
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 3 , September 2020 , pp. 675 - 707
Copyright
© 2020 by Astin Bulletin. All rights reserved

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References

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.CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Cochran, W.G. (1952) The $\chi^{2}$ test of goodness of fit. Annals of Mathematical Statistics, 23, 315–345.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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 Journal 8, 245–255.CrossRefGoogle Scholar
Donoho, D.L. and Johnstone, I.M. (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455.CrossRefGoogle Scholar
Donoho, D.L. and Johnstone, I.M. (1995) Adapting to unknown smoothness via wavelet shrinkage. Journal of the American Statistical Assocation 90, 1200–1224.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
Hyndman, R.J. and Ullah, Md.S. (2007) Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics and Data Analysis 51, 4942–4956.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Lee, R.D. and Carter, L. (1992) Modelling and forecasting the time series of US mortality. Journal of the American Statistical Association 87, 659–671.Google Scholar
Mallat, S.G. (1989a) Multiresolution approximations and wavelet orthonormal bases of $L_{2}(\mathbb{R})$ . Transactions of the American Mathematical Society 315, 69–87.Google Scholar
Mallat, S.G. (1989b) A theory for multiresolution signal decomposition: The wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 674–693.CrossRefGoogle Scholar
Morillas, F., Baeza, I. and Pavia, J.M. (2016) Risk of death: A two-step method using wavelets and piecewise harmonic interpolation. Estadistica Espanola 58, 245–264.Google Scholar
Nickolas, P. (2017) Wavelets: A Student Guide. Cambridge University Press.Google Scholar
Pitacco, E., Denuit, M., Haberman, S. and Olivieri, A. (2009) Modelling Longevity Dynamics for Pensions and Annuity Business. New York: Oxford University Press.Google Scholar
Renshaw, A.E. and Haberman, S. (2003) Lee-Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics 33, 255–272.Google Scholar
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, 556–570.Google Scholar
Renshaw, A.E., Haberman, S. and Hatzoupoulos, P. (1996) The modelling of recent mortality trends in United Kingdom male assured lives. British Actuarial Journal 2, 449–477.CrossRefGoogle Scholar
Strang, G. and Nguyen, T. (1996) Wavelets and Filter Banks. Wellesley, MA.Google Scholar
Tibshirani, R. (1996) Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society – Series B 58, 267–288.Google Scholar
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.pdfGoogle Scholar