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Smoothing dispersed counts with applications to mortality data

Published online by Cambridge University Press:  01 March 2011

V. A. B. Djeundje  and
I. D. Currie
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Abstract

Mortality data are often classified by age at death and year of death. This classification results in a heterogeneous risk set and this can cause problems for the estimation and forecasting of mortality. In the modelling of such data, we replace the classical assumption that the numbers of claims follow the Poisson distribution with the weaker assumption that the numbers of claims have a variance proportional to the mean. The constant of proportionality is known as the dispersion parameter and it enables us to allow for heterogeneity; in the case of insurance data the dispersion parameter also allows for the presence of duplicates in a portfolio. We use both the quasi-likelihood and the extended quasi-likelihood to estimate models for the smoothing and forecasting of mortality tables jointly with smooth estimates of the dispersion parameters. We present three main applications of our method: first, we show how taking account of dispersion reduces the volatility of a forecast of a mortality table; second, we smooth mortality data by amounts, ie, when deaths are amounts claimed and exposed-to-risk are sums assured; third, we present a joint model for mortality by lives and by amounts with the property that forecasts by lives and by amounts are consistent. Our methods are illustrated with data from the Continuous Mortality Investigation.

Keywords

AmountsDuplicatesForecastingHeterogeneityMortalityOver-dispersionP-splinesQuasi-likelihoodSmoothing
Type
Papers
Information
Annals of Actuarial Science , Volume 5 , Issue 1 , March 2011 , pp. 33 - 52
Copyright
Copyright © Institute and Faculty of Actuaries 2010

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References

Bäuerle, N. (1997). Inequalities for stochastic models via supermodular orderings. Communications in Statistics – Stochastic Models, 13, 181–201.CrossRefGoogle Scholar
Beard, R.E., Perks, W. (1949). The relation between the distribution of sickness and the effect of duplicates on the distribution of deaths. Journal of the Institute of Actuaries, 75, 75–86.CrossRefGoogle Scholar
Biatat, V.D., Currie, I.D. (2010). Joint models for classification and comparison of mortality in different countries. In Proceedings of the 25th International Workshop on Statistical Modelling (Ed: A.W. Bowman), 89–94. Glasgow: University of Glasgow.Google Scholar
Breslaw, N.E., Clayton, D.G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, 9–25.Google Scholar
Brouhns, N., Denuit, M., Vermunt, J.K. (2002). A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31, 373–393.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A., 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.CrossRefGoogle Scholar
CMI Committee (1957). Continuous investigation into the mortality of assured lives. Journal of the Institute of Actuaries, 83, 34–36.Google Scholar
CMI Committee (1986). An investigation into the distribution of policies per life assured in the cause of death investigation data. CMIR, 8, 49–58.Google Scholar
Currie, I.D., Durban, M., Eilers, P.H.C. (2004). Smoothing and forecasting mortality rates. Statistical Modelling, 4, 279–298.Google Scholar
Currie, I.D., Durban, M., Eilers, P.H.C. (2006). Generalized linear array models with applications to multidimensional smoothing. Journal of the Royal Statistical Society (Series B), 68, 259–280.CrossRefGoogle Scholar
Daw, R.H. (1946). On the validity of statistical tests of the graduation of a mortality table. Journal of the Institute of Actuaries, 72, 174–190.Google Scholar
Daw, R.H. (1951). Duplicate policies in mortality data. Journal of the Institute of Actuaries, 77, 261–267.CrossRefGoogle Scholar
Denuit, M. (2000). Stochastic analysis of duplicates in life insurance portfolios. German Actuarial Bulletin, 24, 507–514.Google Scholar
Eilers, P.H.C., Marx, B.D. (1996). Flexible smoothing with B-splines and penalties (with Discussion). Statistical Science, 11, 89–121.CrossRefGoogle Scholar
Forfar, D.O., McCutcheon, J.J., Wilkie, A.D. (1988). On graduation by mathematical formula. Journal of the Institute of Actuaries, 115, 1–149.CrossRefGoogle Scholar
Heuer, C. (1997). Modeling of time trends and interactions in vital rates using restricted regression splines. Biometrics, 53, 161–177.Google Scholar
Hinde, J., Demetrio, C.G.B. (1998). Overdispersion: models and estimation. Computational Statistics and Data Analysis, 27, 151–170.Google Scholar
Human Mortality Database (2009). University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (January, 2009).Google Scholar
Li, J.S.H., Hardy, M.R., Tan, K.S. (2009). Uncertainty in mortality forecasting: an extension to the classic Lee-Carter approach. Astin Bulletin, 39, 137–164.CrossRefGoogle Scholar
McCullagh, P., Nelder, J.A. (1989). Generalized Linear Models. London: Chapman and Hall.Google Scholar
Nelder, J.A., Pregibon, D. (1987). An extended quasi-likelihood function. Biometrika, 74, 221–232.Google Scholar
Renshaw, A.E. (1992). Joint modelling for actuarial graduation and duplicate policies. Journal of the Institute of Actuaries, 119, 69–85.Google Scholar
Renshaw, A.E., Hatzopoulos, P. (1996). On the graduation of ‘amounts’. British Actuarial Journal, 2, 185–205.CrossRefGoogle Scholar
Richards, S.J. (2008). Applying survival models to pensioner mortality data. British Actuarial Journal, 14, 257–326.Google Scholar
Richards, S.J., Currie, I.D. (2009). Longevity risk and annuity pricing with the Lee-Carter model. British Actuarial Journal, 15, Part II (to be published).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, 5–61.CrossRefGoogle Scholar
Ruppert, D., Wand, M.P., Carroll, R.J. (2003). Semiparametric regression. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.Google Scholar
Seal, H.L. (1940). Tests of a mortality table graduation. Journal of the Institute of Actuaries, 71, 5–67.Google Scholar
Shaked, M., Shanthikumar, J.G. (1997). Supermodular stochastic orders and positive dependence of random vectors. Journal of Multivariate Analysis, 61, 86–101.Google Scholar
Wedderburn, R.W.M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61, 439–447.Google Scholar
Williams, D.A. (1982). Extra-binomial variation in logistic linear models. Applied Statistics, 31, 144–148.Google 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