Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T07:50:22.341Z Has data issue: false hasContentIssue false
  • English
  • Français

A Bayesian smoothing spline method for mortality modelling

Published online by Cambridge University Press:  15 May 2012

Arto Luoma ,
Anne Puustelli  and
Lasse Koskinen
Arto Luoma*
Affiliation:
University of Tampere, Finland
Anne Puustelli
Affiliation:
University of Tampere, Finland
Lasse Koskinen
Affiliation:
Financial Supervisory Authority of Finland and Helsinki School of Economics, Finland
*
*Correspondence to: Arto Luoma, School of Information Sciences, FIN-33014 University of Tampere, Finland. E-mail: arto.luoma@uta.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a new method for two-dimensional mortality modelling. Our approach smoothes the data set in the dimensions of cohort and age using Bayesian smoothing splines. The method allows the data set to be imbalanced, since more recent cohorts have fewer observations. We suggest an initial model for observed death rates, and an improved model which deals with the numbers of deaths directly. Unobserved death rates are estimated by smoothing the data with a suitable prior distribution. To assess the fit and plausibility of our models we perform model checks by introducing appropriate test quantities. We show that our final model fulfils nearly all requirements set for a good mortality model.

Keywords

Cohort effectForecastingModel checkingParameter uncertaintyStochastic mortality model
Type
Papers
Information
Annals of Actuarial Science , Volume 6 , Issue 2 , 23 August 2012 , pp. 284 - 306
Copyright
Copyright © Institute and Faculty of Actuaries 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Cairns, A.J.G., Blake, D., Dowd, K. (2006a). Pricing death: Frameworks for the valuation and securitization of mortality risk. ASTIN Bulletin, 36, 79–120.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K. (2006b). 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.G., Blake, D., Dowd, K. (2008). Modelling and management of mortality risk: a review. Scandinavian Actuarial Journal, 2, 79–113.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Khalaf-Allah, M. (2011). Bayesian stochastic mortality modelling for two populations. ASTIN Bulletin, 41, 29–59.Google Scholar
Currie, I.D., Durban, M., Eilers, P.H.C. (2004). Smoothing and forecasting mortality rates. Statistical Modelling, 4, 279–298.Google Scholar
Czado, C., Delwarde, A., Denuit, M. (2005). Bayesian Poisson log-linear mortality projections. Insurance: Mathematics and Economics, 36, 260–284.Google Scholar
Dellaportas, P., Smith, A.F.M., Stavropoulos, P. (2001). Bayesian analysis of mortality data. Journal of the Royal Statistical Society. Series A, 164, 275–291.Google Scholar
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. (2004). Bayesian Data Analysis. Chapman & Hall/CRC.Google Scholar
Gelman, A., Rubin, D.B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–511.Google Scholar
Geman, S., Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.Google Scholar
Gilks, W. R., Wild, P. (1992). Applied Statistics, 41, 337–348.Google Scholar
Gilks, W.R., Richardson, S., Spiegelhalter, D., eds. (1996). Markov Chain Monte Carlo in Practice. Chapman & Hall.Google Scholar
Green, P., Silverman, B.W. (1994). Nonparametric Regression and Generalized Linear Models. Chapman & Hall, London.Google Scholar
Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.Google Scholar
Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). www.mortality.orgorwww.humanmortality.de [Accessed April, 2009].Google Scholar
Kogure, A., Kurachi, Y. (2010). A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions. Insurance: Mathematics and Economics, 46, 162–172.Google Scholar
Lee, R.D., Carter, L.R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–675.Google Scholar
Lang, S., Brezger, A. (2004). Bayesian P-Splines. Journal of Computational and Graphical Statistics, 13, 183–212.Google Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E. (1953). Equations of State Calculations by Fast Computing Machines. Journal of Chemical Physics, 21, 1087–1092.Google Scholar
Pedroza, C. (2006). A Bayesian forecasting model: predicting U.S. male mortality. Biostatistics, 7, 530–550.Google Scholar
Plat, R. (2009). On stochastic mortality modeling. Insurance: Mathematics and Economics, 45, 393–404.Google Scholar
Reichmuth, W., Sarferaz, S. (2008). Bayesian demographic modelling and forecasting: An application to US mortality. SFB 649 Discussion paper 2008–052.Google Scholar
Renshaw, A.E., 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
R Development Core Team (2010). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/.Google Scholar
Richards, S.J., Kirkby, J.G., Currie, I.D. (2006). The importance of year of birth in two-dimensional mortality data. British Actuarial Journal, 12, 5–38.Google Scholar
Schmid, V.J., Held, L. (2007). Bayesian age-period-cohort modeling and prediction – BAMP. Journal of Statistical Software, 21, 8. http://www.jstatsoft.org [Accessed April 2012].Google Scholar
Wahba, G. (1978). Improper priors, spline smoothing, and the problem of guarding against model errors in regression. Journal of the Royal Statistical Society B, 40, 364–372.Google Scholar
Wilmoth, J.R., Andreev, K., Jdanov, D., Glei, D.A. (2007). Methods Protocol for the Human Mortality Database. http://www.mortality.org [Accessed April 2009].Google Scholar