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PARSIMONIOUS PARAMETERIZATION OF AGE-PERIOD-COHORT MODELS BY BAYESIAN SHRINKAGE

Published online by Cambridge University Press:  20 September 2017

Gary Venter  and
Şule Şahın
Gary Venter
Affiliation:
School of Risk and Actuarial Studies, Business School, University of New South Wales, Sydney, NSW 2052, Australia Actuarial Sciences Program, School of Professional Studies, Columbia University, New York, NY 10027, USA, E-Mail: gary.venter@gmail.com
Şule Şahın*
Affiliation:
Department of Actuarial Sciences, Hacettepe University, Ankara 06800, Turkey Institute for Financial and Actuarial Mathematics, University of Liverpool, Liverpool, L69 3BX, UK
*
E-Mail: sule@hacettepe.edu.tr
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Abstract

Age-period-cohort models used in life and general insurance can be over-parameterized, and actuaries have used several methods to avoid this, such as cubic splines. Regularization is a statistical approach for avoiding over-parameterization, and it can reduce estimation and predictive variances compared to MLE. In Markov Chain Monte Carlo (MCMC) estimation, regularization is accomplished by the use of mean-zero priors, and the degree of parsimony can be optimized by numerically efficient out-of-sample cross-validation. This provides a consistent framework for comparing a variety of regularized MCMC models, such as those built with cubic splines, linear splines (as ours is), and the limiting case of non-regularized estimation. We apply this to the multiple-trend model of Hunt and Blake (2014).

Keywords

Age-period-cohort modelsmortality trendsMCMCregularizationshrinkage priors
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 89 - 110
Copyright
Copyright © Astin Bulletin 2017 

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Footnotes

*

Şule Şahın's name has been corrected. An erratum notice detailing this change was also published (DOI: 10.1017/asb.2017.39).

References

Antonio, K., Bardoutsos, A. and Ouburg, W. (2015) Bayesian poisson log-bilinear models for mortality projections with multiple populations. European Actuarial Journal, 5, 245–281.CrossRefGoogle Scholar
Barnett, G. and Zehnwirth, B. (2000) Best estimates for reserves. PCAS, 87, 245–303.Google Scholar
Blei, D.M. (2015) Regularized regression. Columbia University, New York, 1–11. http://www.cs.columbia.edu/~blei/fogm/2015F/notes/regularized-regression.pdf.Google Scholar
Cairns, A., Blake, D., Dowd, K., Coughlan, G., 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, 1–35.Google Scholar
Calderhead, B. and Radde, N. (2014) Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems. BMC Bioinformatics, 15, 253, https://bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-15-253.Google Scholar
Carlson, E. (2008) The Lucky Few. Netherlands: Springer.Google Scholar
Casella, G. and George, E.I. (1992) Explaining the Gibbs sampler. The American Statistician, 46 (3), 167–174.Google Scholar
Chung, Fung, M., Peters, G. and Shevchenko, P. (2016) A unified approach to mortality modelling using state-space framework: characterisation, identification, estimation and forecasting. Working Paper. arXiv:1605.09484v1.Google Scholar
Efron, B. and Morris, C. (1975) Data analysis using Stein's estimator and its generalizations. Journal of the American Statistical Association, 70 (350), 311–319.CrossRefGoogle Scholar
Fienberg, S.E. and Mason, W.M. (1978) Identification and estimation of age-period-cohort models in the analysis of discrete archival data. Sociological Methodology, 10, 1–67.CrossRefGoogle Scholar
Frost, W.H. (1939) The age selection of mortality from tuberculosis in successive decades. American Journal of Hygiene, 30 (3A), 91–96.Google Scholar
Gelfand, A.E. (1996) Model determination using sampling-based methods. In Markov Chain Monte Carlo in Practice (eds. Gilks, W.R., Richardson, S. and Spiegelhalter, D.J.), pp. 145–162. London: Chapman and Hall.Google Scholar
Gelfand, A.E., Dey, D.K. and Chang, H. (1992) Model determination using predictive distributions with implementation via sampling-based methods. Technical Report No. 462 for the Office of Naval Research, Department of Statistics, Stanford University.Google Scholar
Greenberg, B.G., Wright, J.J. and Sheps, C.G. (1950) A technique for analyzing some factors affecting the incidence of syphilis. Journal of the American Statistical Association, 45 (251), 373–399.Google Scholar
Hachemeister, C.A. and Stanard, J.N. (1975) IBNR claims count estimation with static lag functions. ASTIN Colloquium, Portimão, Portugal.Google Scholar
Hastie, T., Tibshirani, R. and Wainwright, M. (2015) Statistical Learning with Sparsity. Boca Raton, FL: CRC Press.Google Scholar
Hoerl, A. and Kennard, R. (1970) Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12, 55–67.Google Scholar
Hoffman, M.D. and Gelman, A. (2014) Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems. Journal of Machine Learning Research, 15, 1351–1381.Google Scholar
Hunt, A. and Blake, D. (2014) A general procedure for constructing mortality models. North American Actuarial Journal, 18, 116–138.Google Scholar
Keisler (2000) Elementary Calculus: An Infinitesimal Approach. Prindle, Weber & Schmidt, http://www.math.wisc.edu/~keisler/calc.html.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2008) Loss models: From Data to Decisions, 3rd edition. New York: Wiley.CrossRefGoogle Scholar
Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Associationl, 87, 659–675.Google Scholar
McDonald, J. (1984) Some generalized functions for the size distribution of income. Econometrica, 52, 647–663.CrossRefGoogle Scholar
Ntzoufras, I. (2010) Lesson 1 An introduction to MCMC sampling methods. https://www.statistics.com/papers/LESSON1_Notes_MCMC.pdf.Google Scholar
van Ravenzwaaij, D., Cassey, P. and Brown, S. (2016) A simple introduction to Markov Chain Monte Carlo sampling. Psychonomic Bulletin & Review, 2016, 1–12. https://link.springer.com/article/10.3758%2Fs13423-016-1015-8.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
Ryder, N.B. (1965) The cohort as a concept in the study of social change. American Sociological Review, 30 (6), 843–861.Google Scholar
Taylor, G. (1977) Separation of inflation and other effects from the distribution of non-life insurance claims delays. ASTIN Bulletin, 9, 217–230.Google Scholar
Vehtari, A., Gelman, A. and Gabry, J. (2017) Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27 (5), 1413–1432.Google Scholar
Venter, G.G. (1983) Transformed beta and gamma distributions and aggregate losses. Proceedings of the Casualty Actuarial Society, LXX, 156–193.Google Scholar
Verbeek, H.G. (1972) An approach to the analysis of claims experience in excess of loss reinsurance. ASTIN Bulletin, 6, 195–202.Google Scholar
Wolfram (2016) Gamma function. http://mathworld.wolfram.com/GammaFunction.html.Google Scholar
Xu, Y., Sherris, M. and Ziveyi, J. (2015) The application of affine processes in multi-cohort mortality model. University of New South Wales Business School Research Paper No. 2015ACTL13.CrossRefGoogle Scholar
Ye, J. (1998) On measuring and correcting the effects of data mining and model selection. Journal of the American Statistical Association, 93, 120–131.Google Scholar