×

Bayesian mortality forecasting with overdispersion. (English) Zbl 1406.62130

Summary: The ability to produce accurate mortality forecasts, accompanied by a set of representative uncertainty bands, is crucial in the planning of public retirement funds and various life-related businesses. In this paper, we focus on one of the drawbacks of the Poisson Lee-Carter model [N. Brouhns et al., Insur. Math. Econ. 31, No. 3, 373–393 (2002; Zbl 1074.62524)] that imposes mean-variance equality, restricting mortality variations across individuals. Specifically, we present two models to potentially account for overdispersion. We propose to fit these models within the Bayesian framework for various advantages, but primarily for coherency. Markov Chain Monte Carlo (MCMC) methods are implemented to carry out parameter estimation. Several comparisons are made with the Bayesian Poisson Lee-Carter model [C. Czado et al., ibid. 36, No. 3, 260–284 (2005; Zbl 1110.62142)] to highlight the importance of accounting for overdispersion. We demonstrate that the methodology we developed prevents over-fitting and yields better calibrated prediction intervals for the purpose of mortality projections. Bridge sampling is used to approximate the marginal likelihood of each candidate model to compare the models quantitatively.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62M20 Inference from stochastic processes and prediction
62N05 Reliability and life testing
91D20 Mathematical geography and demography
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Abel, G. J., Fanplot: an R package for visualizing sequential distributions, R JOURNAL, 7, 1, 15-23, (2015)
[2] Alho, J. M., Modelling and forecasting the time series of U.S. mortality, J. Amer. Statist. Assoc., 87, 673-674, (1992)
[3] Alzaid, A.; Sultan, K. S., Discriminating between gamma and lognormal distributions with applications, J. King Saud Univ.-Sci., 21, 2, 99-108, (2009)
[4] Antonio, K.; Bardoutsos, A.; Ouburg, W., Bayesian Poisson log-bilinear models for mortality projections with multiple populations, Eur. Actuar. J., 5, 2, 245-281, (2015) · Zbl 1329.91111
[5] Atkinson, A. C., A method for discriminating between models, J. R. Stat. Soc. Ser. B Stat. Methodol., 32, 3, 323-353, (1970) · Zbl 0225.62020
[6] Berger, J. O., Statistical decision theory and Bayesian analysis, (2013), Springer Science & Business Media
[7] Booth, H.; Tickle, L., Mortality modelling and forecasting: A review of methods, Ann. Actuar. Sci., 3, 1-2, 3-43, (2008)
[8] Brouhns, N.; Denuit, M.; Keilegom, I. V., Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scand. Actuar. J., 2005, 3, 212-224, (2005) · Zbl 1092.91038
[9] Brouhns, N.; Denuit, M.; Vermunt, J. K., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance Math. Econom., 31, 3, 373-393, (2002) · Zbl 1074.62524
[10] Brown, J. R., Redistribution and insurance: mandatory annuitization with mortality heterogeneity, J. Risk Insurance, 70, 1, 17-41, (2003)
[11] Cairns, A.; Blake, D.; Dowd, K., A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, J. Risk Insurance, 73, 4, 687-718, (2006)
[12] Cho, H. K.; Bowman, K. P.; North, G. R., A comparison of gamma and lognormal distributions for characterizing satellite rain rates from the tropical rainfall measuring mission, J. Appl. Meteorol., 43, 11, 1586-1597, (2004)
[13] Cox, D. R., Tests of separate families of hypotheses, (Proceedings of the Fourth Berkeley Symposium in Mathematical Statistics and Probability, (1961), University of California Press Berkeley), 105-123 · Zbl 0201.52102
[14] Cox, D. R., Further results on tests of separate families of hypotheses, J. R. Stat. Soc. Ser. B Stat. Methodol., 24, 2, 406-424, (1962) · Zbl 0131.35801
[15] Czado, C.; Delwarde, A.; Denuit, M., Bayesian Poisson log-bilinear mortality projections, Insurance Math. Econom., 36, 3, 260-284, (2005) · Zbl 1110.62142
[16] Delwarde, A.; Denuit, M.; Partrat, C., Negative binomial version of the Lee-Carter model for mortality forecasting, Appl. Stoch. Models Bus. Ind., 23, 5, 381-401, (2007) · Zbl 1150.91426
[17] Dick, E. J., Beyond ‘lognormal versus gamma’: discrimination among error distributions for generalized linear models, Fish. Res., 70, 2-3, 351-366, (2004)
[18] Firth, D., Multiplicative errors:log-normal or gamma?, J. R. Stat. Soc. Ser. B Stat. Methodol., 50, 2, 266-268, (1988)
[19] Gelfand, A. E.; Sahu, S. K., Identifiability, improper priors and Gibbs sampling for generalized linear models, J. Amer. Statist. Assoc., 94, 445, 515-533, (1999) · Zbl 1072.62611
[20] Gelman, A., Prior distributions for variance parameters in hierarchical models, Bayesian Anal., 1, 3, 515-533, (2006) · Zbl 1331.62139
[21] Gelman, A.; Rubin, D. B., Inference from iterative simulation using multiple sequences, Statist. Sci., 7, 4, 457-511, (1992) · Zbl 1386.65060
[22] Gelman, A.; Rubin, D. B.; Carlin, J. B.; Stern, H. S., Bayesian Data Analysis, (1995), Chapman and Hall Ltd
[23] Girosi, F.; King, G., Demographic Forecasting, (2008), Princeton University Press
[24] Kass, R. E.; Raftery, A. E., Bayes factors, J. Amer. Statist. Assoc., 90, 430, 773-795, (1995) · Zbl 0846.62028
[25] Lee, R. D.; Carter, L. R., Modelling and forecasting U.S. mortality, J. Amer. Statist. Assoc., 87, 419, 659-671, (1992) · Zbl 1351.62186
[26] Lee, R. D.; Miller, T., Evaluating the performance of the Lee-Carter method for forecasting mortality, Demography, 38, 4, 537-549, (2001)
[27] Li, J., An application of MCMC simulation in mortality projection for populations with limited data, Demography, 30, 1, 1-48, (2014)
[28] Li, S. H.; Hardy, M. R.; Tan, K. S., Uncertainty in mortality forecasting: an extension to the classical Lee-Carter approach, ASTIN Bull., 39, 1, 137-164, (2009) · Zbl 1203.91113
[29] Lunn, D.; Jackson, C.; Best, N.; Thomas, A.; Spiegelhalter, D., The BUGS Book: A Practical Introduction to Bayesian Analysis, (2013), Chapman and Hall/CRC · Zbl 1281.62009
[30] Meng, X. L.; Wong, W. H., Simulating ratios of normalizing constants via a simple identity: a theoretical exploration, Statist. Sinica, 6, 4, 831-860, (1996) · Zbl 0857.62017
[31] O’Hagan, A.; Forster, J., Kendall’s Advanced Theory of Statistics, Vol. 2B, (2004), Kendall’s Library of Statistics · Zbl 1058.62002
[32] Pedroza, C., A Bayesian forecasting model: predicting U.S. male mortality, Biostatistics, 7, 4, 530-550, (2006) · Zbl 1170.62397
[33] Raftery, A. E.; Chunn, J. L., Bayesian probabilistic projections of life expectancy for all countries, Demography, 50, 3, 777-801, (2013)
[34] Renshaw, A. E.; Haberman, H., A cohort-based extension to the Lee-Carter model for mortality reduction factors, Insurance Math. Econom., 38, 3, 556-570, (2005) · Zbl 1168.91418
[35] Roberts, G. O.; Rosenthal, J. S., Optimal scaling for various metropolis-Hastings algorithms, Statist. Sci., 16, 4, 351-367, (2001) · Zbl 1127.65305
[36] Roberts, G. O.; Sahu, S. K., Updating schemes, correlation structure, blocking and parameterization for the Gibbs sampler, J. Roy. Statist. Soc. Ser. B (Methodol.), 59, 2, 291-317, (1997) · Zbl 0886.62083
[37] Tuljapurkar, S.; Li, N.; Boe, C., A universal pattern of mortality decline in the g7 countries, Lett. Nat., 405, 789-792, (2000)
[38] Wiens, B. L., When log-normal and gamma models give different results: a case study, Amer. Statist., 53, 2, 89-93, (1999)
[39] Wiśniowski, A.; Smith, P. W.F.; Bijak, J.; Forster, J. J.; Raymer, J., Bayesian population forecasting: extending the Lee-Carter method, Demography, 52, 3, 1035-1059, (2015)
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.