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Modelling Adult Mortality in Small Populations: The Saint Model

Published online by Cambridge University Press:  09 August 2013

Søren Fiig Jarner  and
Esben Masotti Kryger
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Abstract

The mortality evolution of small populations often exhibits substantial variability and irregular improvement patterns making it hard to identify underlying trends and produce plausible projections. We propose a methodology for robust forecasting based on the existence of a larger reference population sharing the same long-term trend as the population of interest. The reference population is used to estimate the parameters in a frailty model for the underlying intensity surface. A multivariate time series model describing the deviations of the small population mortality from the underlying mortality is then fitted and forecasted. Coherent long-term forecasts are ensured by the underlying frailty model while the size and variability of short- to medium-term deviations are quantified by the time series model. The frailty model is particularly well suited to describe the changing improvement patterns in old age mortality. We apply the method to Danish mortality data with a pooled international data set as reference population.

Keywords

SAINT modelsmall populationadult mortalityfrailtyDanish and international mortality
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 2 , November 2011 , pp. 377 - 418
Copyright
Copyright © International Actuarial Association 2011

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References

Aalen, O.O. (1988) Heterogeneity in survival analysis. Statistics in Medicine 7, 1121–1137.CrossRefGoogle ScholarPubMed
Abbring, J.H. and van der Berg, G.J. (2007) The unobserved heterogeneity distribution in duration analysis. Biometrika 94, 87–99.CrossRefGoogle Scholar
Andreev, K.F. (2002) Evolution of the Danish Population from 1835 to 2000. Monographs on Population Aging, 9. Odense University Press.Google Scholar
Barbi, E. (2003) Assessing the rate of ageing of the human population. Working Paper. Max Planck Institute for Demographic Research.Google Scholar
Biatat, V.D. and Currie, I.D. (2010) Joint models for classification and comparison of mortality in different countries. Proceedings of 25rd International Workshop on Statistical Modelling, Glasgow, 89–94.Google Scholar
Bongaarts, J. (2005) Long-range trends in adult mortality: Models and projection methods. Demography 42, 23–49.Google Scholar
Booth, H., Hyndman, R.J., Tickle, L. and de Jong, P. (2006) Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research 15, 289–310.Google Scholar
Booth, H., Maindonald, J. and Smith, L. (2002) Applying Lee-Carter under conditions of varying mortality decline. Population Studies 56, 325–336.Google Scholar
Brockwell, P.J. and Davis, R.A. (1991) Time Series: Theory and Methods, Second Edition. Springer-Verlag, New York.Google 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
Butt, Z. and Haberman, S. (2004) Application of frailty-based mortality models using generalized linear models. ASTIN Bulletin 34, 175–197.Google Scholar
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, 687–718.Google Scholar
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 & Wales and the United States. North American Actuarial Journal 13, 1–35.CrossRefGoogle Scholar
Cairns, A.J.G. (2000) A discussion of parameter and model uncertainty in insurance. Insurance: Mathematics and Economics 27, 313–330.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D. and Khalaf-Allah, M. (2011) Bayesian stochastic mortality modelling for two populations. ASTIN Bulletin 41, 29–59.Google Scholar
Currie, I.D., Durban, M. and Eilers, P.H.C. (2004) Smoothing and forecasting mortality rates. Statistical Modelling 4, 279–298.Google Scholar
de Jong, P. and Tickle, L. (2006) Extending Lee-Carter mortality forecasting. Mathematical Population Studies 13, 1–18.CrossRefGoogle Scholar
Dowd, K., Cairns, A.J.G. and Blake, D. (2006) Mortality-dependent financial risk measures. Insurance: Mathematics and Economics 38, 427–440.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2010a) Backtesting stochastic mortality models: An ex-post evaluation of multi-period-ahead density forecasts. North American Actuarial Journal 14, 281–298.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2010b) Evaluating the goodness of fit of stochastic mortality models. Insurance: Mathematics and Economics 47, 255–265.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011) A gravity model of mortality rates for two related populations. North American Actuarial Journal 15, 334–356.Google Scholar
Efron, B. and Tibshirani, R.J. (1993) An Introduction to the Bootstrap. Chapman & Hall, New York.CrossRefGoogle Scholar
Gavrilov, L.A. and Gavrilova, N.S. (1991) The Biology of Life Span: A Quantitative Approach, ed. V.P. Skulachev. Harwood Academic Publishers, Chur.Google Scholar
Hougaard, P. (1984) Life table methods for heterogeneous populations: Distributions describing the heterogeneity. Biometrika 71, 75–83.Google Scholar
Jarner, S.F., Kryger, E.M., and Dengsøe, C. (2008) The evolution of death rates and life expectancy in Denmark. Scandinavian Actuarial Journal 108, 147–173.Google Scholar
Koissi, M.-C., Shapiro, A.F. and Högnäs, G. (2006) Evaluating and extending the Lee-Carter model for mortality forecasting: Bootstrap confidence intervals. Insurance: Mathematics and Economics 38, 1–20.Google Scholar
Lee, R.D. and Carter, L.R. (1992) Modeling and Forecasting of U.S. Mortality. Journal of the American Statistical Association 87, 659–675.Google Scholar
Lee, R.D. and Miller, T. (2001) Evaluating the performance of the Lee- Carter method for forecasting mortality. Demography 38, 537–549.Google Scholar
Li, J. S.-H. and Hardy, M.R. (2009) Measuring basis risk involved in longevity hedges. Working paper, University of Waterloo.Google Scholar
Li, J. S.-H., Hardy, M.R. and Tan, K.S. (2009) Uncertainty in mortality forecasting: An extension to the classic Lee-Carter approach. ASTIN Bulletin 39, 137–164.Google Scholar
Li, N. and Lee, R. (2005) Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method. Demography 42, 575–594.Google Scholar
Li, N., Lee, R. and Tuljapurkar, S. (2004) Using the Lee-Carter method to forecast mortality for populations with limited data. International Statistical Review 72, 19–36.CrossRefGoogle Scholar
Nusselder, W.J. and Mackenbach, J.P. (2000) Lack of improvement of life expectancy at advanced ages in The Netherlands. International Journal of Epidemiology 29, 140–148.CrossRefGoogle ScholarPubMed
Olivieri, A. (2006) Heterogeneity in survival models. applications to pensions and life annuities. Belgian Actuarial Journal 6, 23–39.Google Scholar
Olivieri, A. and Pitacco, E. (2009) Stochastic mortality: the impact on target capital. ASTIN Bulletin 39, 541–564.CrossRefGoogle Scholar
Pitacco, E., Denuit, M., Haberman, S. and Olivieri, A. (2009) Modelling longevity dynamics for pensions and annuity business. Oxford University Press, Oxford.Google Scholar
Plat, R. (2009) Stochastic portfolio specific mortality and the quantification of mortality basis risk. Insurance: Mathematics and Economics 45, 123–132.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
Richards, S.J., Kirkby, J.G. and Currie, I.D. (2006) The importance of year of birth in two-dimensional mortality data. British Actuarial Journal 12, 5–38.Google Scholar
Thatcher, A.R. (1999) The Long-Term Pattern of Adult Mortality and the Highest Attained Age. Journal of the Royal Statistical Society. Series A 162, 5–43.CrossRefGoogle ScholarPubMed
Tuljapurkar, S., Li, N. and Boe, C. (2000) A universal pattern of mortality decline in the G7 countries. Nature 405, 789–792.Google Scholar
Vaupel, J.W. (1999) Discussion on “The Long-Term Pattern of Adult Mortality and the Highest Attained Age” by A.R. Thatcher. Journal of the Royal Statistical Society. Series A 162, 31–32.Google Scholar
Vaupel, J.W., Manton, K.G. and Stallard, E. (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography 16, 439–454.Google Scholar
Wang, S.S. and Brown, R.L. (1998) A frailty model for projection of human mortality improvements. Journal of Actuarial Practise 6, 221–241.Google Scholar
Wienke, A. (2010) Frailty models in survival analysis. Chapman & Hall.CrossRefGoogle Scholar
Willets, R.C. (2004) The cohort effect: Insights and explanations. British Actuarial Journal 10, 833–877.Google Scholar
Wilmoth, J.R. (1998) Is the pace of Japanese mortality decline converging toward international trends? Population and Development Review 24, 593–600.CrossRefGoogle Scholar
Wilson, C. (2001) On the scale of global demographic convergence 1950-2000. Population and Development Review 27, 155–171.Google Scholar
Yashin, A.I. and Iachine, I.A. (1997) How frailty models can be used for evaluating longevity limits: Taking advantage of an interdisciplinary approach. Demography 34, 31–48.Google Scholar