## Robust forecasting of mortality and fertility rates: a functional data approach.(English)Zbl 1162.62434

Summary: A new method is proposed for forecasting age-specific mortality and fertility rates observed over time. This approach allows for smooth functions of age, is robust for outlying years due to wars and epidemics, and provides a modelling framework that is easily adapted to allow for constraints and other information. Ideas from functional data analysis, nonparametric smoothing and robust statistics are combined to form a methodology that is widely applicable to any functional time series data observed discretely and possibly with error. The model is a generalization of the Lee-Carter (LC) model commonly used in mortality and fertility forecasting. The methodology is applied to French mortality data and Australian fertility data, and the forecasts obtained are shown to be superior to those from the LC method and several of its variants.

### MSC:

 62P10 Applications of statistics to biology and medical sciences; meta analysis 62M20 Inference from stochastic processes and prediction 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G08 Nonparametric regression and quantile regression

### Software:

SemiPar; Human Mortality; fda (R); COBS
Full Text:

### References:

 [1] Bell, W.R., Comparing and assessing time series methods for forecasting age-specific fertility and mortality rates, J. official statist., 13, 3, 279-303, (1997) [2] Bell, W.R., Monsell, B., 1991. Using principal components in time series modelling and forecasting of age-specific mortality rates. In: Proceedings of the American Statistical Association, Social Statistics Section, pp. 154-159. [3] Booth, H.; Maindonald, J.; Smith, L., Applying lee – carter under conditions of variable mortality decline, Population studies, 56, 3, 325-336, (2002) [4] Booth, H.; Tickle, L.; Smith, L., Evaluation of the variants of the lee – carter method of forecasting mortality: a multi-country comparison, New Zealand population rev., 31, 1, 13-34, (2005) [5] Booth, H., Hyndman, R.J., Tickle, L., de Jong, P., 2006. Lee-Carter mortality forecasting: a multi-country comparison of variants and extensions. $$\operatorname{Demographic}$$$$\operatorname{Research}$$ 15, to appear. [6] Bozik, J.E., Bell, W.R., 1987. Forecasting age-specific fertility using principal components. In: Proceedings of the American Statistical Association, Social Statistics Section. San Francisco, CA, pp. 396-401. [7] Carter, L.R.; Lee, R.D., Modelling and forecasting US sex differentials in mortality, Internat. J. forecasting, 8, 3, 393-411, (1992) [8] Chen, C.; Liu, L.-M., Joint estimation of model parameters and outlier effects in time series, J. amer. statist. assoc., 88, 284-297, (1993) · Zbl 0775.62229 [9] Croux, C.; Ruiz-Gazen, A., A fast algorithm for robust principal components based on projection pursuit, (), 211-216 · Zbl 0900.62300 [10] Croux, C.; Ruiz-Gazen, A., High breakdown estimators for principal components: the projection-pursuit approach revisited, J. multivariate anal., 95, 1, 206-226, (2005) · Zbl 1065.62040 [11] Currie, I.D.; Durban, M.; Eilers, P.H.C., Smoothing and forecasting mortality rates, Statist. model., 4, 4, 279-298, (2004) · Zbl 1061.62171 [12] Dauxois, J.; Pousse, A.; Romain, Y., Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference, J. multivariate anal., 12, 136-154, (1982) · Zbl 0539.62064 [13] De Jong, P.; Tickle, L., Extending lee – carter mortality forecasting, Math. population studies, 13, 1, 1-18, (2006) · Zbl 1151.91742 [14] Erbas, B., Hyndman, R.J., Gertig, D.M., 2006. Forecasting age-specific breast cancer mortality using functional data models. Statist. Med., to appear. [15] Ferraty, F.; Vieu, P., Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination, Nonparametric statist., 16, 1-2, 111-125, (2004) · Zbl 1049.62039 [16] He, X.; Ng, P., COBS: qualitatively constrained smoothing via linear programming, Comput. statist., 14, 315-337, (1999) · Zbl 0941.62037 [17] Hössjer, O.; Croux, C., Generalizing univariate signed rank statistics for testing and estimating a multivariate location parameter, Nonparametric statist., 4, 293-308, (1995) · Zbl 1381.62113 [18] Hubert, M.; Rousseeuw, P.J.; Verboven, S., A fast method of robust principal components with applications to chemometrics, Chemometrics and intelligent laboratory systems, 60, 101-111, (2002) [19] Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at $$\langle$$www.mortality.org⟩ (data downloaded on 1 May 2006). [20] Jolliffe, I.T., Principal component analysis, (2002), Springer New York · Zbl 1011.62064 [21] Lee, R.D., Modeling and forecasting the time series of U.S. fertility: age distribution, range, and ultimate level, Internat. J. forecasting, 9, 187-202, (1993) [22] Lee, R.D.; Carter, L.R., Modeling and forecasting U.S. mortality, J. amer. statist. assoc., 87, 659-675, (1992) · Zbl 1351.62186 [23] Lee, R.D.; Miller, T., Evaluating the performance of the lee – carter method for forecasting mortality, Demography, 38, 4, 537-549, (2001) [24] Lee, R.D.; Carter, L.R.; Tuljapurkar, S., Disaggregation in population forecasting: do we need it? and how to do it simply?, Math. population studies, 5, 3, 217-234, (1995) · Zbl 0900.92192 [25] Li, G.; Chen, Z., Projection-pursuit approach to robust dispersion matrices and principal components: primary theory and Monte Carlo, J. amer. statist. assoc., 80, 391, 759-766, (1985) · Zbl 0595.62060 [26] Li, S.-H.; Chan, W.-S., Outlier analysis and mortality forecasting: the united kingdom and Scandinavian countries, Scand. actuar. J., 3, 187-211, (2005) · Zbl 1092.91050 [27] Locantore, N.; Marron, J.S.; Simpson, D.G.; Tripoli, N.; Zhang, J.T.; Cohen, K.L., Robust principal component analysis for functional data, Sociedad de estadística e investigación operativa test, 8, 1, 1-73, (1999) · Zbl 0980.62049 [28] Ramsay, J.O.; Dalzell, C.J., Some tools for functional data analysis, J. roy. statist. soc., ser. B, 53, 3, 539-572, (1991) · Zbl 0800.62314 [29] Ramsay, J.O.; Silverman, B.W., Functional data analysis, (2005), Springer New York · Zbl 1079.62006 [30] Renshaw, A.E.; Haberman, S., Lee – carter mortality forecasting: a parallel generalized linear modelling approach for england and wales mortality projections, Appl. statist., 52, 1, 119-137, (2003) · Zbl 1111.62359 [31] Ruppert, D.; Wand, M.P.; Carroll, R.J., Semiparametric regression, (2003), Cambridge University Press New York · Zbl 1038.62042 [32] Silverman, B.W., Smoothed functional principal components analysis by choice of norm, Ann. statist., 24, 1-24, (1996) · Zbl 0853.62044 [33] Simonoff, J.S., Smoothing methods in statistics, (1996), Springer New York · Zbl 0859.62035 [34] Valderrama, M.J., Ocaña, F.A., Aguilera, A.M., 2002. Forecasting PC-ARIMA models for functional data. In: Härdle, W., Rönz, B. (Eds.), Proceedings in Computational Statistics, pp. 25-36. [35] Wilmoth, J.R., 2002. Methods Protocol for the Human Mortality Database. Revised 1 October 2002. Downloaded on 18 July 2003. $$\langle$$http://www.mortality.org/Public/Docs/MethodsProtocol.pdf⟩. [36] Wolf, D.A., 2004. Another variation on the Lee-Carter model. Paper Presented at the Annual Meeting of the Population Association of America, April 2004. [37] Wood, S.N., Monotonic smoothing splines fitted by cross validation, SIAM J. sci. comput., 15, 5, 1126-1133, (1994) · Zbl 0821.65002 [38] Wood, S.N., Thin plate regression splines, J. roy. statist. soc., ser. B, 65, 1, 95-114, (2003) · Zbl 1063.62059
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.