×

A parameterized approach to modeling and forecasting mortality. (English) Zbl 1156.91394

Summary: A new method is proposed of constructing mortality forecasts. This parameterized approach utilizes Generalized Linear Models (GLMs), based on heteroscedastic Poisson (non-additive) error structures, and using an orthonormal polynomial design matrix. Principal Component (PC) analysis is then applied to the cross-sectional fitted parameters. The produced model can be viewed either as a one-factor parameterized model where the time series are the fitted parameters, or as a principal component model, namely a log-bilinear hierarchical statistical association model of L. A. Goodman [Measures, models, and graphical displays in the analysis of cross-classified data. J. Am. Statist. Assoc. 86, No. 416, 1085–1111 (1991)] or equivalently as a generalized Lee-Carter model with \(p\) interaction terms. Mortality forecasts are obtained by applying dynamic linear regression models to the PCs. Two applications are presented: Sweden (1751-2006) and Greece (1957-2006).

MSC:

91B30 Risk theory, insurance (MSC2010)

Software:

Human Mortality
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Bell, W.R., Comparing and assessing time series methods for forecasting age-specific fertility and mortality rates, Journal of official statistics, 13, 3, 279-303, (1997)
[2] Bell, W.R., Monsell, B.C., 1991. Using principal components in time series modelling and forecasting of age-specific mortality rates. In: Proceedings of the American Statistical Association, Social Statistics Section, pp. 154-159
[3] Booth, H.; Hyndman, R.J.; Tickle, L., Lee – carter mortality forecasting: A multicountry comparison of variants and extensions, Demographic research, 15, 9, 289-310, (2006)
[4] Booth, H.; Maindonald, J.; Smith, L., Applying lee – carter under conditions of variable mortality decline, Population studies, 56, 325-336, (2002)
[5] Bozik, J.E., Bell, W.R., 1987. Forecasting age specific fertility using principal components. In: Proceedings of the American Statistical Association, Social Statistics Section, pp. 396-401
[6] Brillinger, D.R., The natural variability of vital rates and associated statistics, Biometrics, 42, 4, 693-734, (1986) · Zbl 0611.62136
[7] Brouhns, N.; Denuit, M.; Vermunt, J.K., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance: mathematics and economics, 31, 3, 373-393, (2002) · Zbl 1074.62524
[8] Cox, D.R., Renewal theory, (1967), Methuen London · Zbl 0168.16106
[9] Cox, D.R., Some remarks on overdispersion, Biometrika, 70, 1, 269-274, (1983) · Zbl 0511.62007
[10] Cramer, H.; Wold, H., Mortality variations in Sweden. A study in graduation and forecasting, Skandinavisk actuarie tidskrift, 18, 161-206, (1935) · Zbl 0012.36306
[11] Currie, I.D.; Durban, M.; Eilers, P.H.C., Smoothing and forecasting mortality rates, Statistical. modelling., 4, 4, 279-298, (2004) · Zbl 1061.62171
[12] De Jong, P.; Tickle, L., Extending lee – carter mortality forecasting, Mathematical population studies, 13, 1, 1-18, (2006) · Zbl 1151.91742
[13] De Moivre, A., 1725. Annuities on Lives: Or, the valuation of annuities upon any number of lives, as also, of reversions. London. Quoted in Smith, D. and Keyfitz, N. (1977), Mathematical demography: Selected Readings, Biomathematics, vol. 6, Springer-Verlag, New York, 1977
[14] Efron, B.; Tibshirani, R., An introduction to the bootstrap, (1998), CRC Press Boca Raton, FL
[15] Forfar, D.O.; McCutcheon, J.J.; Wilkie, A.D., On graduation by mathematical formula, Journal of institute of actuaries, 115, 1, 1-135, (1988)
[16] General Secretariat of National Statistical Service of Greece. Available at: http://www.statistics.gr/ (data downloaded at June, 2008)
[17] Gompertz, B., On the nature of the law of human mortality and on a new method of determining the value of life contingencies, Philosophical transactions of royal society, 115, 513-585, (1825)
[18] Goodman, L.A., Measures, models, and graphical displays in the analysis of cross-classified data, Journal of the American statistical association, 86, 416, 1085-1111, (1991) · Zbl 0850.62093
[19] Hagnell, M., A multivariate time series analysis of fertility, adult mortality, nuptiality and real wages in Sweden 1751-1850: A comparison of some different approaches, Journal of official statistics, 7, 437-455, (1991)
[20] Harvey, A., Forecasting, structural time series models and the Kalman filter, (1991), Cambridge University Press
[21] Heligman, L.; Pollard, J.H., The age pattern of mortality, Journal of the institute of actuaries, 107, 1, 49-80, (1980)
[22] Human Mortality Database, University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at:http://www.mortality.org (data downloaded at June, 2008)
[23] Hyndman, R.J., Ullah, M.S., 2005. Robust forecasting of mortality and fertility rates: A functional data approach. Working Paper, Department of Economics and Business Statistics, Monash University.http://www.robhyndman.info/papers/funcfor.htm · Zbl 1162.62434
[24] Kendall, M.G.; Stuart, A., The advanced theory of statistics, vol. 2, (1967), Griffin London
[25] Keyfitz, N., Experiments in the projection of mortality, Canadian studies in population, 18, 2, 1-17, (1991)
[26] Koissi, M.-C.; Shapiro, A.F.; Hognas, G., Evaluating and extending the lee – carter model for mortality forecasting: bootstrap confidence interval, Insurance: mathematics and economics, 38, 1, 1-20, (2006) · Zbl 1098.62138
[27] Ledermann, S.; Breas, J., LES dimensions de la mortalite, Population, 14, 637-682, (1959)
[28] Lee, R.D.; Carter, L.R., Modelling and forecasting US mortality, Journal of the American statistical association, 87, 659-671, (1992)
[29] Lee, R.D., Modelling and forecasting the time series of US fertility: age patterns, range, and ultimate level, International journal of forecasting, 9, 187-202, (1993)
[30] Lee, R.D., The lee – carter method for forecasting mortality, with various extensions and applications, North American actuarial journal, 4, 1, 80-93, (2000) · Zbl 1083.62535
[31] Lee, R.D.; Miller, T., Evaluating the performance of the lee – carter method for forecasting mortality, Demography, 38, 4, 537-549, (2001)
[32] Mardia, K.V.; Kent, J.T.; Bibby, J.M., Multivariate analysis, (1997), Academic Press · Zbl 0432.62029
[33] McNown, R.; Rogers, A., Forecasting mortality: A parameterized time series approach, Demography, 26, 4, 645-660, (1989)
[34] McNown, R.; Rogers, A., Forecasting cause-specific mortality using time series methods, International journal of forecasting, 8, 413-432, (1992)
[35] Newbold, P., Bos, T., 1985. Stochastic parameter regression models. Series: Quantitative Applications in the Social Sciences. Sage University Papers. Series/Number 07-051
[36] Pitacco, E., Survival models in dynamic context: A survey, Insurance: mathematics & economics, 35, 2, 279-298, (2004) · Zbl 1079.91050
[37] Pitacco, E.; Denuit, M.; Haberman, S.; Olivieri, A.-M., Modelling longevity dynamics for pensions and annuity business, (2009), Oxford University Press Oxford · Zbl 1163.91005
[38] Renshaw, A.E., Actuarial graduation practice and generalised linear and non-linear models, Journal of the institute of actuaries, 118, 2, 295-312, (1991)
[39] Renshaw, A.E., Joint modelling for actuarial graduation and duplicate policies, Journal of the institute of actuaries, 119, 1, 69-85, (1992)
[40] Renshaw, A.E.; Haberman, S.; Hatzopoulos, P., The modelling of recent mortality trends in united kingdom male assured lives, British actuarial journal, 2, 2, 449-477, (1996)
[41] Renshaw, A.E.; Haberman, S.; Hatzopoulos, P., On the duality of assumptions underpinning the construction of life tables, Astin bulletin, 27, 1, 5-22, (1997)
[42] Renshaw, A.E.; Haberman, S., Lee – carter mortality forecasting with age-specific enhancement, Insurance: mathematics and economics, 33, 2, 255-272, (2003) · Zbl 1103.91371
[43] Renshaw, A.E.; Haberman, S., On the forecasting of mortality reduction factors, Insurance: mathematics and economics, 32, 3, 379-401, (2003) · Zbl 1025.62041
[44] Renshaw, A.E.; Haberman, S., A cohort-based extension to the lee – carter model for mortality reduction factors, Insurance: mathematics and economics, 38, 3, 556-570, (2006) · Zbl 1168.91418
[45] Renshaw, A.E.; Haberman, S., On simulation-based approaches to risk measurement in mortality with specific to Poisson Lee-Carter modelling, Insurance: mathematics and economics, 42, 3, 797-816, (2008) · Zbl 1152.91598
[46] Sinamurthy, M., 1987. Principal components representation of ASFR: Model of fertility estimation and projection. CDC research monograph number, 16. Cairo Demographic Center. pp. 655-693
[47] Sithole, T.; Haberman, S.; Verrall, R.J., An investigation into parametric models for mortality projections, with applications to immediate annuitants’ and life office pensioners’ data, Insurance: mathematics & economics, 27, 3, 285-312, (2000) · Zbl 1055.62555
[48] ()
[49] Taylor, C.J., 2007. Engineering Department, Lancaster University, Lancaster, LA1 4YR, United Kingdom. Web http://www.es.lancs.ac.uk/cres/captain/
[50] Tuljapurkar, S.; Li, N.; Boe, C., A universal pattern of mortality decline in the G7 countries, Nature, 405, 789-792, (2000)
[51] Wilmoth, J.R., Demography of longevity: past, present, and future trends, Experimental gerontology, 35, 1111-1129, (2000)
[52] Yue, J.C., Yang, S.S., Huang, H.-C., 2008. A study of the Lee-Carter model with a jump. Living to 100 Symposium January 7-9, 2008. Featured Papers
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.