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GAUSSIAN PROCESS MODELS FOR MORTALITY RATES AND IMPROVEMENT FACTORS

Published online by Cambridge University Press:  22 August 2018

Mike Ludkovski ,
Jimmy Risk  and
Howard Zail
Mike Ludkovski
Affiliation:
Department of Statistics & Applied Probability, University of California, Santa Barbara CA 93106-3110, USA E-Mail: ludkovski@pstat.ucsb.edu
Jimmy Risk*
Affiliation:
Department of Mathematical Sciences, Cal Poly Pomona, Pomona, CA 91768, USA
Howard Zail
Affiliation:
Actuary and Founder of Elucidor, LLC, NY 10016, USA E-Mail: hzail@elucidor.com
*
E-Mail: jrisk@cpp.edu
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Abstract

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We develop a Gaussian process (GP) framework for modeling mortality rates and mortality improvement factors. GP regression is a nonparametric, data-driven approach for determining the spatial dependence in mortality rates and jointly smoothing raw rates across dimensions, such as calendar year and age. The GP model quantifies uncertainty associated with smoothed historical experience and generates full stochastic trajectories for out-of-sample forecasts. Our framework is well suited for updating projections when newly available data arrives, and for dealing with “edge” issues where credibility is lower. We present a detailed analysis of GP model performance for US mortality experience based on the CDC (Center for Disease Control) datasets. We investigate the interaction between mean and residual modeling, Bayesian and non-Bayesian GP methodologies, accuracy of in-sample and out-of-sample forecasting, and stability of model parameters. We also document the general decline, along with strong age-dependency, in mortality improvement factors over the past few years, contrasting our findings with the Society of Actuaries (SOA) MP-2014 and -2015 models that do not fully reflect these recent trends.

Keywords

mortalitymortality improvementmortality modelingGaussian processes

Information

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 3 , September 2018 , pp. 1307 - 1347
Copyright
Copyright © Astin Bulletin 2018 

References

Adler, R.J. (1981) The Geometry of Random Fields, Classics in Applied Mathematics, volume 62. SIAM. Chichester, NY.Google Scholar
Brooks, S., Gelman, A., Jones, G. and Meng, X.-L. (2011) Handbook of Markov Chain Monte Carlo. CRC press. Boca Raton, FL.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 (3), 373–393.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 (4), 687–718.Google Scholar
Cairns, A.J., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011) Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48 (3), 355–367.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 and Wales and the United States. North American Actuarial Journal, 13 (1), 1–35.Google Scholar
Camarda, C.G. (2012) Mortalitysmooth: An R package for smoothing Poisson counts with P-splines. Journal of Statistical Software, 50 (1), 1–24.Google Scholar
Carpenter, B., Lee, D., Brubaker, M.A., Riddell, A., Gelman, A., Goodrich, B., Guo, J., Hoffman, M., Betancourt, M. and Li, P. (2016) Stan: A probabilistic programming language. Journal of Statistical Software, 76 (1). doi 10.18637/jss.v076.i01Google Scholar
Continuous Mortality Investigation (2015) The CMI mortality projections model, CMI 2015. Technical report, CMI Working Paper 84.Google Scholar
Cressie, N. (2015) Statistics for Spatial Data. Canada: John Wiley & Sons.Google Scholar
Currie, I.D. (2013) Smoothing constrained generalized linear models with an application to the Lee-Carter model. Statistical Modelling, 13 (1), 69–93.Google Scholar
Currie, I.D. (2016) On fitting generalized linear and non-linear models of mortality. Scandinavian Actuarial Journal, 2016 (4), 356–383.Google Scholar
Currie, I.D., Durban, M. and Eilers, P.H. (2004) Smoothing and forecasting mortality rates. Statistical Modelling, 4 (4), 279–298.Google Scholar
Czado, C., Delwarde, A. and Denuit, M. (2005) Bayesian poisson log-bilinear mortality projections. Insurance: Mathematics and Economics, 36 (3), 260–284.Google Scholar
Debón, A., Martínez-Ruiz, F. and Montes, F. (2010) A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities. Insurance: Mathematics and Economics, 47 (3), 327–336.Google Scholar
Delwarde, A., Denuit, M. and Eilers, P. (2007) Smoothing the Lee–Carter and Poisson log-bilinear models for mortality forecasting a penalized log-likelihood approach. Statistical Modelling, 7 (1), 29–48.Google Scholar
Dokumentov, A. and Hyndman, R.J. (2014) Bivariate Data with Ridges: Two-Dimensional Smoothing of Mortality Rates. Technical report, Working paper series, Monash University.Google Scholar
Girosi, F. and King, G. (2008) Demographic Forecasting. Princeton, NJ: Princeton University Press.Google Scholar
Gramacy, R. and Taddy, M. (2012) Tgp, an R package for treed Gaussian process models. Journal of Statistical Software, 33, 1–48.Google Scholar
Hunt, A. and Blake, D. (2014) A general procedure for constructing mortality models. North American Actuarial Journal, 18 (1), 116–138.Google Scholar
Hyndman, R.J. and Ullah, M.S. (2007) Robust forecasting of mortality and fertility rates: a functional data approach. Computational Statistics & Data Analysis, 51 (10), 4942–4956.Google Scholar
Lee, M.R. and Owen, A.B. (2015) Single nugget kriging. Technical report, arXiv preprint arXiv:1507.05128.Google Scholar
Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting US mortality. Journal of the American Statistical Association, 87 (419), 659–671.Google Scholar
Li, H. and O'Hare, C. (2015) Mortality forecast: Local or global? Technical report, Available at SSRN 2612420.Google Scholar
Ludkovski, M. (2015) Kriging metamodels for bermudan option pricing. arXiv:1509.02179.Google Scholar
Mitchell, D., Brockett, P., Mendoza-Arriaga, R. and Muthuraman, K. (2013) Modeling and forecasting mortality rates. Insurance: Mathematics and Economics, 52 (2), 275–285.Google Scholar
Picheny, V. and Ginsbourger, D. (2013) A nonstationary space-time Gaussian process model for partially converged simulations. SIAM/ASA Journal on Uncertainty Quantification, 1 (1), 57–78.Google Scholar
Purushotham, M., Valdez, E. and Wu, H. (2011) Global mortality improvement experience and projection techniques. Technical report, Society of Actuaries.Google Scholar
Renshaw, A., Haberman, S. and Hatzopoulos, P. (1996) The modelling of recent mortality trends in united kingdom male assured lives. British Actuarial Journal, 2 (2), 449–477.Google Scholar
Renshaw, A.E. and Haberman, S. (2003) Lee–Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics, 33 (2), 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 (3), 556–570.Google Scholar
Riihimäki, J. and Vehtari, A. (2010) Gaussian processes with monotonicity information. Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, in PMLR, 9:645–652.Google Scholar
Rosner, B., Raham, C., Orduña, F., Chan, M., Xue, L., Zak, B. and Yang, G. (2013) Literature review and assessment of mortality improvement rates in the US population: Past experience and future long-term trends. Technical report, Society of Actuaries.Google Scholar
Roustant, O., et al. (2012) Dicekriging, Diceoptim: Two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. Journal of Statistical Software, 51 (1), 1–55.Google Scholar
Salemi, P., Staum, J. and Nelson, B.L. (2013) Generalized integrated Brownian fields for simulation metamodeling. In Proceedings of the 2013 Winter Simulation Conference, IEEE Press, pp. 543–554.Google Scholar
Sithole, T.Z., Haberman, S. and Verrall, R.J. (2000) An investigation into parametric models for mortality projections, with applications to immediate annuitants and life office pensioners data. Insurance: Mathematics and Economics, 27 (3), 285–312.Google Scholar
SOA (2014a) Mortality improvement scale MP-2014 report. Technical report, Retirement Plans Experience Committee. https://www.soa.org/Research/Experience-Study/Pension/research-2014-mp.aspx.Google Scholar
SOA (2014b) RP-2014 mortality tables. Technical report, Society of Actuaries Pension Experience Study. https://www.soa.org/Research/Experience-Study/pension/research-2014-rp.aspx.Google Scholar
SOA (2015) Mortality improvement scale MP-2015. Technical report, Retirement Plans Experience Committee. https://www.soa.org/Research/Experience-Study/Pension/research-2015-mp.aspx.Google Scholar
Villegas, A.M., Kaishev, V.K. and Millossovich, P. (2015) StMoMo: An R package for stochastic mortality modelling. Technical report, SSRN documentation at papers.ssrn.com/sol3/papers.cfm?abstract-id=2698729.Google Scholar
Whittaker, E.T. (1922) On a new method of graduation. Proceedings of the Edinburgh Mathematical Society, 41, 63–75.Google Scholar
Williams, C.K. and Rasmussen, C.E. (2006) Gaussian Processes for Machine Learning. Massachusetts: The MIT Press.Google Scholar
Wilmoth, J.R. and Shkolnikov, V. (2010) Human Mortality Database. University of California.Google Scholar
Wu, R. (2016) Gaussian Process and Functional Data Methods for Mortality Modelling. PhD thesis, Department of Mathematics University of Leicester.Google Scholar
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