## Smoothing Poisson common factor model for projecting mortality jointly for both sexes.(English)Zbl 1390.91204

Summary: We consider a modification to the Poisson common factor model and utilise a generalised linear model (GLM) framework that incorporates a smoothing process and a set of linear constraints. We extend the standard GLM model structure to adopt Lagrange methods and $$P$$-splines such that smoothing and constraints are applied simultaneously as the parameters are estimated. Our results on Australian, Canadian and Norwegian data show that this modification results in an improvement in mortality projection in terms of producing more accurate forecasts in the out-of-sample testing. At the same time, projected male-to-female ratio of death rates at each age converges to a constant and the residuals of the models are sufficiently random, indicating that the use of smoothing does not adversely affect the fit of the model. Further, the irregular patterns in the estimates of the age-specific parameters are moderated as a result of smoothing and this model can be used to produce more regular projected life tables for pricing purposes.

### MSC:

 91B30 Risk theory, insurance (MSC2010) 62P05 Applications of statistics to actuarial sciences and financial mathematics 91D20 Mathematical geography and demography

gamair
Full Text:

### References:

 [1] Booth, H.; Hyndman, R.; Tickle, L.; De Jong, P., Lee-Carter mortality forecasting: a multi-country comparison of variants and extensions, Demographic Research, 15, 289-310, (2006) [2] Booth, H.; Maindonald, J.; Smith, L., (2001) [3] Booth, H.; Maindonald, J.; Smith, L., Applying Lee-Carter under conditions of variable mortality decline, Population Studies, 56, 325-336, (2002) [4] Booth, H.; Tickle, L., Mortality modelling and forecasting: a review of methods, Annals of Actuarial Science, 3, 3-43, (2008) [5] Brouhns, N.; Denuit, M.; Van Keilegom, I., Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scandinavian Actuarial Journal, 3, 212-224, (2005) · Zbl 1092.91038 [6] Brouhns, N.; Denuit, M.; Vermunt, J., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance: Mathematics and Economics, 31, 373-393, (2002) · Zbl 1074.62524 [7] Cairns, A. J.G.; Blake, D.; Dowd, K.; Coughlan, G. D.; Khalaf-Allah, M., Bayesian stochastic mortality modelling for two populations, ASTIN Bulletin, 41, 29-59, (2011) [8] Chan, W. S.; Li, J. S.H.; Li, J., The CBD mortality indexes: modeling and applications, North American Actuarial Journal, 18, 35-58, (2014) [9] Currie, I., Smoothing constrained generalized linear models with an application to the Lee-Carter model, Statistical Modelling, 13, 69-93, (2013) [10] Currie, I.; Durban, M.; Eilers, P., Smoothing and forecasting mortality rates, Statistical Modelling, 4, 279-298, (2004) · Zbl 1061.62171 [11] Currie, I., On fitting generalized linear and non-linear models of mortality, Scandinavian Actuarial Journal, 4, 356-383, (2016) · Zbl 1401.91123 [12] De Jong, P.; Tickle, L., Extending Lee-Carter mortality forecasting, Mathematical Population Studies, 13, 1-18, (2006) · Zbl 1151.91742 [13] Delwarde, A.; Denuit, M.; Eilers, P., Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting: a penalized log-likelihood approach, Statistical Modelling, 7, 29-48, (2007) [14] Eilers, P.; Marx, B., Flexible smoothing with B-splines and penalties, Statistical Science, 11, 89-121, (1996) · Zbl 0955.62562 [15] Eilers, P.; Marx, B., Splines, knots, and penalties, Wiley Interdisciplinary Reviews: Computational Statistics, 2, 637-653, (2010) [16] Hastie, T.; Tibshirani, R., Generalized Additive Models, (1990), London: Chapman and Hall, London · Zbl 0747.62061 [17] Hyndman, R. J.; Booth, H.; Yasmeen, F., Coherent mortality forecasting: the product-ratio method with functional time series models, Demography, 50, 261-283, (2013) [18] Hyndman, R.; Dokumentov, A., Two-dimensional smoothing of mortality rates, (2013) [19] Hyndman, R.; Shahid Ullah, M., Robust forecasting of mortality and fertility rates: A functional data approach, Computational Statistics & Data Analysis, 51, 4942-4956, (2007) · Zbl 1162.62434 [20] Jarner, S.; Kryger, E., Modelling adult mortality in small populations: the SAINT model, ASTIN Bulletin, 41, 377-418, (2011) · Zbl 1239.91128 [21] Kogure, A.; Li, J.; Kamiya, S., A Bayesian multivariate risk-neutral method for pricing reverse mortgages, North American Actuarial Journal, 18, 242-257, (2014) [22] Koissi, M.; Shapiro, A., Fuzzy formulation of the Lee-Carter model for mortality forecasting, Insurance: Mathematics and Economics, 39, 287-309, (2006) · Zbl 1151.91576 [23] Koissi, M.; Shapiro, A.; Högnäs, G., Evaluating and extending the Lee-Carter model for mortality forecasting: Bootstrap confidence interval, Insurance: Mathematics and Economics, 38, 1-20, (2006) · Zbl 1098.62138 [24] Lee, R.; Carter, L., Modelling and forecasting U.S. mortality, Journal of the American Statistical Association, 87, 659-671, (1992) · Zbl 1351.62186 [25] Li, J., Projections of New Zealand mortality using the Lee-Carter model and its augmented common factor extension, New Zealand Population Review, 36, 27-53, (2010) [26] Li, J., A Poisson common factor model for projecting mortality and life expectancy jointly for females and males, Population Studies, 67, 111-126, (2013) [27] Li, J., A quantitative comparison of simulation strategies for mortality projection, Annals of Actuarial Science, 8, 281-297, (2014) [28] Li, J., An application of MCMC simulation in mortality projection for populations with limited data, Demographic Research, 30, 1-48, (2014) [29] Li, N.; Lee, R., Coherent mortality forecasts for a group of populations: an extension of the Lee-Carter method, Demography, 42, 575-594, (2005) [30] Mccullagh, P.; Nelder, J., Generalized Linear Models, (1989), London: Chapman & Hall, London · Zbl 0744.62098 [31] Parr, N.; Li, J.; Tickle, L., A cost of living longer: projections of the effects of prospective mortality improvement on economic support ratios for 14 advanced economies, Population Studies, 70, 181-200, (2016) [32] Renshaw, A.; Haberman, S., Lee-Carter mortality forecasting: a parallel generalized linear modelling approach for England and Wales mortality projections, Applied Statistics, 52, 119-137, (2003) · Zbl 1111.62359 [33] Renshaw, A.; Haberman, S., A cohort-based extension to the Lee-Carter model for mortality reduction factors, Insurance: Mathematics and Economics, 38, 556-570, (2006) · Zbl 1168.91418 [34] Renshaw, A. E.; Haberman, S., On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling, Insurance: Mathematics and Economics, 42, 797-816, (2008) · Zbl 1152.91598 [35] Renshaw, A.; Haberman, S.; Hatzopoulos, P., The modelling of recent mortality trends in United Kingdom male assured lives, British Actuarial Journal, 2, 449-477, (1996) [36] Tan, C. I.; Li, J.; Li, J. S.H.; Balasooriya, U., Parametric mortality indexes: from index construction to hedging strategies, Insurance: Mathematics and Economics, 59, 285-299, (2014) · Zbl 1306.91140 [37] Tuljapurkar, S.; Li, N.; Boe, C., A universal pattern of mortality decline in the G7 countries, Nature, 405, 789-792, (2000) [38] Wilmoth, J. R., (1993) [39] Wood, S. N., Generalized Additive Models: An Introduction with R, (2006), London: Chapman & Hall/CRC, London · Zbl 1087.62082 [40] Yang, B.; Li, J.; Balasooriya, U., Cohort extensions of the Poisson common factor model for modelling both genders jointly, Scandinavian Actuarial Journal, 2, 93-112, (2016) · Zbl 1401.91203
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.