1. bookVolume 37 (2021): Issue 3 (September 2021)
    Special Issue on Population Statistics for the 21st Century
Journal Details
First Published
01 Oct 2013
Publication timeframe
4 times per year
access type Open Access

Modelling Frontier Mortality Using Bayesian Generalised Additive Models

Published Online: 13 Sep 2021
Volume & Issue: Volume 37 (2021) - Issue 3 (September 2021)<br/>Special Issue on Population Statistics for the 21st Century
Page range: 569 - 589
Received: 01 Nov 2019
Accepted: 01 Nov 2020
Journal Details
First Published
01 Oct 2013
Publication timeframe
4 times per year

Mortality rates differ across countries and years, and the country with the lowest observed mortality has changed over time. However, the classic Science paper by Oeppen and Vaupel (2002) identified a persistent linear trend over time in maximum national life expectancy. In this article, we look to exploit similar regularities in age-specific mortality by considering for any given year a hypothetical mortality ‘frontier’, which we define as the lower limit of the force of mortality at each age across all countries. Change in this frontier reflects incremental advances across the wide range of social, institutional and scientific dimensions that influence mortality. We jointly estimate frontier mortality as well as mortality rates for individual countries. Generalised additive models are used to estimate a smooth set of baseline frontier mortality rates and mortality improvements, and country-level mortality is modelled as a set of smooth, positive deviations from this, forcing the mortality estimates for individual countries to lie above the frontier. This model is fitted to data for a selection of countries from the Human Mortality Database (2019). The efficacy of the model in forecasting over a ten-year horizon is compared to a similar model fitted to each country separately.


Aburto, J.M., F. Villavicencio, U. Basellini, S. Kjærgaard, and J.W. Vaupel. 2020. “Dynamics of life expectancy and life span equality.” Proceedings of the National Academy of Sciences of the United States of America 117 (10): 5250–9. DOI: https://doi.org/10.1073/pnas.1915884117.10.1073/pnas.1915884117707189432094193 Search in Google Scholar

Alho, J.M. 2019. “Forecasting Life Expectancy: A Statistical Look at Model Choice and Use of Auxiliary Series.” In Old and New Perspectives on Mortality Forecasting, edited by Tommy Bengtsson and Nico Keilman, 185–95. Springer Open. DOI: https://doi.org/10.1007/978-3-030-05075-7_15.10.1007/978-3-030-05075-7_15 Search in Google Scholar

Bengtsson, T. 2019. “Linear Increase in Life Expectancy: Past and Present.” In Old and New Perspectives on Mortality Forecasting, edited by Tommy Bengtsson and Nico Keilman, 221–34. Springer Open. DOI: https://doi.org/10.1007/978-3-030-05075-7_17.10.1007/978-3-030-05075-7_17 Search in Google Scholar

Bergeron-Boucher, M.P., V. Canudas-Romo, J. Oeppen, and J.W. Vaupel. 2017. “Coherent forecasts of mortality with compositional data analysis.” Demographic Research 37 (1): 527–66. DOI: https://doi.org/10.4054/DemRes.2017. Search in Google Scholar

Bergeron-Boucher, M.P., V. Canudas-Romo, M. Pascariu, and R. Lindahl-Jacobsen. 2018. “Modeling and forecasting sex differences in mortality: a sex-ratio approach.” Genus 74 (1). DOI: https://doi.org/10.1186/s41118-018-0044-8.10.1186/s41118-018-0044-8628085030595608 Search in Google Scholar

Biatat, V.D., and I.D. Currie. 2010. “Joint models for classification and comparison of mortality in different countries.” In Proceedings of 25rd International Workshop on Statistical Modelling: 89–94. 5–9 July 2010, Glasgow, UK. Available at: http://www.statmod.org/workshops_archive_proceedings_2010.htm (accessed October 2020). Search in Google Scholar

Bijak, J., D. Kupiszewska, M. Kupiszewski, K. Saczuk, and A. Kicinger. 2007. “Population and labour force projections for 27 European countries, 2002–2052: Impact of international migration on population ageing.” European Journal of Population 23 (1): 1–31. DOI: https://doi.org/10.1007/s10680-006-9110-6.10.1007/s10680-006-9110-6279805020076759 Search in Google Scholar

Bohk-Ewald, C., and R. Rau. 2017. “Probabilistic mortality forecasting with varying agespecific survival improvements.” Genus 73 (1): 1–37. DOI: https://doi.org/10.1186/s41118-016-0017-8.10.1186/s41118-016-0017-8523374628133393 Search in Google Scholar

Booth, H., R. Hyndman, L. Tickle, and P. de Jong. 2006. “Lee-Carter mortality forecasting: a multi-country comparison of variants and extensions.” Demographic Research 15 (9): 289–310. DOI: https://doi.org/10.4054/DemRes.2006. Search in Google Scholar

Booth, H., and L. Tickle. 2008. “Mortality Modelling and Forecasting: A Review of Methods.” Annals of Acturial Science 3 (1–2): 3–43. DOI: https://doi.org/10.1017/S1748499500000440.10.1017/S1748499500000440 Search in Google Scholar

Cairns, A.J.G., D. Blake, K. Dowd, G.D. Coughlan, D. Epstein, A. Ong, I. Balevich, D. Coughlan, D. Epstein, and A. Ong. 2009. “A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States.” North American Actuarial Journal 13 (March): 1–35. DOI: https://doi.org/10.1080/10920277.2009.10597538.10.1080/10920277.2009.10597538 Search in Google Scholar

Cairns, A.J.G., D. Blake, K. Dowd, G.D. Coughlan, and M. Khalaf-Allah. 2011. “Bayesian stochastic mortality modelling for two populations.” ASTIN Bulletin 41 (1): 29–59. DOI: https://doi.org/10.2143/AST.41.L2084385. Search in Google Scholar

Case, A., and A. Deaton. 2017. “Mortality and morbidity in the 21st century.” Brookings Papers on Economic Activity 2017 (Spring): 397–476. DOI: https://doi.org/10.1353/eca.2017.0005.10.1353/eca.2017.0005564026729033460 Search in Google Scholar

Currie, I.D., M. Durban, and P.H.C. Eilers. 2004. “Smoothing and forecasting mortality rates.” Statistical Modelling 4: 279–98. DOI: https://doi.org/10.1191/1471082x04st080oa.10.1191/1471082X04st080oa Search in Google Scholar

Enchev, V., T. Kleinow, and A.J.G. Cairns. 2017. “Multi-population mortality models: fitting, forecasting and comparisons.” Scandinavian Actuarial Journal 2017 (4): 319–42. DOI: https://doi.org/10.1080/03461238.2015.1133450.10.1080/03461238.2015.1133450 Search in Google Scholar

Hilton, J., E. Dodd, J. Forster, and P.W.F. Smith. 2019. “Projecting UK Mortality using Bayesian Generalised Additive Models.” Journal of the Royal Statistical Society. Series C 68 (1): 29–49. DOI: https://doi.org/10.1111/rssc.12299.10.1111/rssc.12299 Search in Google Scholar

Human Mortality Database. 2019. “Human Mortality Database.” University of California, Berkeley (USA); Max Planck Institute for Demographic Research (Germany). Available at: DOI: http://www.mortality.org/cgi-bin/hmd. (accessed in October 2019). Search in Google Scholar

Hyndman, R.J., H. Booth, and F. Yasmeen. 2013. “Coherent Mortality Forecasting: The Product-Ratio Method with Functional Time Series Models.” Demography 50 (1): 261–83. DOI: https://doi.org/10.1007/s13524-012-0145-5.10.1007/s13524-012-0145-523055234 Search in Google Scholar

Hyndman, R.J., and Md. Shahid Ullah. 2007. “Robust forecasting of mortality and fertility rates: A functional data approach.” Computational Statistics and Data Analysis 51 (10): 4942–56. DOI: https://doi.org/10.1016/jxsda.2006.07.028. Search in Google Scholar

Keyfitz, N. 1977. “What difference would it make if cancer were eradicated? An examination of the taeuber paradox.” Demography 14 (4): 411–18. DOI: https://doi.org/10.2307/2060587.10.2307/2060587 Search in Google Scholar

Kleinow, T. 2015. “A common age effect model for the mortality of multiple populations.” Insurance: Mathematics and Economics 63. Elsevier B.V.: 147–52. DOI: https://doi.org/10.1016/j.insmatheco.2015. Search in Google Scholar

Lang, S., and A. Brezger. 2004. “Bayesian P-Splines.” Journal of Computational and Graphical Statistics 13 (1): 183–212. DOI: https://doi.org/10.1198/1061860043010.10.1198/1061860043010 Search in Google Scholar

Lee, R.D., and S. Tuljapurkar. 1994. “Stochastic population forecasts for the United States: beyond high, medium, and low.” Journal of the American Statistical Association 89 (428): 1, 1175–89. DOI: http://www.jstor.org/stable/10.2307/2290980. Search in Google Scholar

Lee, R. 2019. “Mortality Forecasts and Linear Life Expectancy Trends.” In Old and New Perspectives on Mortality Forecasting, edited by Tommy Bengtsson and Nico Keilman, 167–83. Springer Open. DOI: https://doi.org/10.1007/978-3-030-05075-7_14.10.1007/978-3-030-05075-7_14 Search in Google Scholar

Lee, R.D, and L.R. Carter. 1992. “Modeling and Forecasting U.S Mortality.” Journal of the American Statistical Association 87 (419): 659–71. Available at: http://www.jstor.org/stable/2290201.10.2307/2290201 Search in Google Scholar

Li, N., and R. Lee. 2005. “Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method.” Demography 42 (3): 575–94. DOI: https://doi.org/10.1353/dem.2005.0021.10.1353/dem.2005.0021135652516235614 Search in Google Scholar

Li, N., R. Lee, and P. Gerland. 2013. “Extending the Lee-Carter method to model the rotation of age patterns of mortality decline for long-term projection.” Demography 50 (6): 2037–51. DOI: https://doi.org/10.1007/s13524-013-0232-2.10.1007/s13524-013-0232-2455058923904392 Search in Google Scholar

Luy, M. 2003. “Causes of Male Excess Mortality: Insights from Cloistered Populations.” Population and Development Review 29 (4). Blackwell Publishing Ltd.: 647–76. DOI: doi.org/10.1111/j.1728-4457.2003.00647.x.10.1111/j.1728-4457.2003.00647.x Search in Google Scholar

Oeppen, J. 2019. “Life Expectancy Convergence Among Nations Since 1820: Separating the Effects of Technology and Income.” In Old and New Perspectives on Mortality Forecasting, edited by Tommy Bengtsson and Nico Keilman, 197–219. Springer Open. DOI: https://doi.org/10.1007/978-3-030-05075-7_16.10.1007/978-3-030-05075-7_16 Search in Google Scholar

Oeppen, J., and J.W. Vaupel. 2002. “Broken Limits to Life Expectancy.” Science 296 (5570): 1029–31. DOI: https://doi.org/10.1126/science.1069675.10.1126/science.106967512004104 Search in Google Scholar

Olshansky, S.J., B.A. Carnes, and A. Desesquelles. 2001. “Prospects for Human Longevity.” Science 291 (5508): 1492–2. DOI: https://doi.org/10.1126/science.291.5508.1491.10.1126/science.291.5508.149111234076 Search in Google Scholar

Pascariu, M.D., V. Canudas-Romo, and J.W. Vaupel. 2018. “The doublegap life expectancy forecasting model.” Insurance: Mathematics and Economics 78 (2018): 339–50. DOI: https://doi.org/10.1016/j.insmatheco.2017. Search in Google Scholar

Raftery, A.E., J.L. Chunn, P. Gerland, and H. Sevcikova. 2013. “Bayesian Probabilistic Projections of Life Expectancy for All Countries.” Demography 50 (3): 777–801. DOI: https://doi.org/10.1007/s13524-012-0193-x.10.1007/s13524-012-0193-x390428923494599 Search in Google Scholar

Schinzinger, E., M.M. Denuit, and M.C. Christiansen. 2016. “A multivariate evolutionary credibility model for mortality improvement rates.” Insurance: Mathematics and Economics 69: 70–81. DOI: https://doi.org/10.1016Zj.insmatheco.2016. Search in Google Scholar

Stan Development Team. 2019. “Stan Modeling Language Users Guide and Reference Manual.” Available at: DOI: http://mc-stan.org/index.html (accessed in October 2019). Search in Google Scholar

Torri, T., and J.W. Vaupel. 2012. “Forecasting life expectancy in an international context.” International Journal of Forecasting 28 (2): 519–31. DOI: https://doi.org/10.1016/j.ijforecast.2011. Search in Google Scholar

Tuljapurkar, S., N. Li, and C. Boe. 2000. “A universal pattern of mortality decline in the G7 countries.” Nature 405 (6788): 789–92. DOI: https://doi.org/10.1038/35015561.10.1038/3501556110866199 Search in Google Scholar

Vallin, J., and F. Mesle. 2009. “The segmented trend line of highest life expectancies.” Population and Development Review 35 (1): 159–87. DOI: https://doi.org/10.1111/j.1728-457.2009.00264.x. Search in Google Scholar

Vaupel, J.W. 1986. “How Change in Age-Specific Mortality Affects Life Expectancy.” Population Studies 40 (1): 147–57. DOI: https://doi.org/10.1080/0032472031000141896.10.1080/003247203100014189611611920 Search in Google Scholar

Vaupel, J.W., and V. Canudas Romo. 2003. “Decomposing change in life expectancy: A bouquet of formulas in honor of Nathan Keyfitz’s 90th birthday.” Demography 40 (2): 201–16. DOI: https://doi.org/10.2307/3180798.10.2307/3180798 Search in Google Scholar

Vaupel, J.W., and V. Canudas Romo. 2000. “How Mortality Improvement Increases Population Growth.” In Optimization, Dynamics, and Economic Analysis: Essays in Honour of Gustav Feichtinger, edited by Engelbert J Dockner, Richard F Hartl, Mikulas Luptacik, and Gerhard Sorger, 345–52. Berlin/Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-642-57684-3_29.10.1007/978-3-642-57684-3_29 Search in Google Scholar

Wachter, K.W. 1997. “Between Zeus and the Salmon: Introduction.” In Between Zeus and the Salmon, edited by Kenneth W Wachter and Caleb E Finch. Washington DC: National Academy Press. Search in Google Scholar

Wachter, K.W., D. Steinsaltz, and S.N. Evans. 2014. “Evolutionary shaping of demographic schedules.” Proceedings of the National Academy of Sciences of the United States of America 111 (SUPPL.3): 10846–53. DOI: https://doi.org/10.1073/pnas.1400841111.10.1073/pnas.1400841111411392525024186 Search in Google Scholar

Wilson, C. 2001. “On the scale of global demographic convergence, 1950–2000.” Population and Development Review 27 (1): 155–71. DOI: https://doi.org/10.1111/j.1728-4457.2001.00155.x.10.1111/j.1728-4457.2001.00155.x18589488 Search in Google Scholar

Wood, S.N. 2006. Generalised Additive Models: An Introduction with R. Boca Raton: Chapman; Hall / CRC Press. Search in Google Scholar

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