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The stratified sampling bootstrap for measuring the uncertainty in mortality forecasts. (English) Zbl 1362.62182

Summary: In this paper, we propose a procedure for reducing the uncertainty in mortality projections, on the basis of a log bilinear Poisson Lee-Carter model [A. Renshaw and S. Haberman, J. R. Stat. Soc., Ser. C, Appl. Stat. 52, No. 1, 119–137 (2003; Zbl 1111.62359)]. In the literature, because the non-linear nature of the quantities under consideration has prevented analytical solutions, simulation techniques have been used in order to provide prediction intervals for forecasted quantities (for example, N. Brouhns et al. [Scand. Actuar. J. 2005, No. 3, 212–224 (2005; Zbl 1092.91038)], A. Renshaw and S. Haberman, Insur. Math. Econ. 42, No. 2, 797–816 (2008; Zbl 1152.91598)]. In this respect, we adopt the bootstrap simulation approach in order to measure the uncertainty affecting mortality projections. In particular, we propose making the bootstrap procedure more efficient by using a specific variance reducing technique, the so-called stratified sampling technique. To this end, we propose a two stage simulation bootstrap procedure where variance reducing techniques are combined with the simple bootstrap of the Poisson Lee-Carter version. Numerical applications are shown using the results for some datasets.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62D05 Sampling theory, sample surveys
62F40 Bootstrap, jackknife and other resampling methods
62M20 Inference from stochastic processes and prediction
91B30 Risk theory, insurance (MSC2010)
91B82 Statistical methods; economic indices and measures
92B15 General biostatistics
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References:

[1] Booth H, Maindonald J, Smith L (2002) Applying Lee-Carter under conditions of variable mortality decline. Popul Stud 56:325–336
[2] Brillinger DR (1986) The natural variability of vital rates and associated statistics. Biometrics 42:693–734 · Zbl 0611.62136
[3] Brouhns N, Denuit M, Vermunt JK (2002a) A Poisson log-bilinear approach to the construction of projected lifetables. Insur Math Econ 31:373–393 · Zbl 1074.62524
[4] Brouhns N, Denuit M, Vermunt JK (2002b) Measuring the longevity risk in mortality projections. Bull Swiss Assoc Actuaries 2:105–130 · Zbl 1187.62158
[5] Brouhns N, Denuit M, van Keilegom I (2005) Bootstrapping the Poisson log-bilinear model for mortality forecasting. Scand Actuar J 3:212–224 · Zbl 1092.91038
[6] D’Amato V, Haberman S, Russolillo M (2009) Efficient bootstrap applied to the Poisson Log-Bilinear Lee Carter Model. Applied Stochastic Models and Data Analysis–ASMDA 2009 Selected Papers, ISBN 978-9955-28-463-5
[7] Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman & Hall, New York & London · Zbl 0835.62038
[8] England P, Verrall R (1999) Analytic and bootstrap estimate of prediction errors in claims reserving. Insur Math Econ 25:281–293 · Zbl 0944.62093
[9] Hoedemakers T, Beirlant J, Goovaerts M, Dhaene J (2003) Confidence bounds for discounted loss reserves. Insur Math Econ 33:297–316 · Zbl 1103.91367
[10] Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (data downloaded on [date])
[11] Hyndman RJ, Ullah S (2005) Robust forecasting of mortality and fertility rates: a functional data approach. Working paper, Department of Econometrics and Business Statistics, Monash University. http://www-personal.buseco.monash.edu.au/\(\sim\)hyndman/papers/funcfor.htm
[12] Koissi MC, Shapiro AF, Hognas G (2006) Evaluating and extending the Lee–Carter model for mortality forecasting: bootstrap confidence interval. Insur Math Econ 26:1–20 · Zbl 1098.62138
[13] Lee RD, Carter LR (1992) Modelling and forecasting U.S. mortality. J Am Stat Assoc 87:659–671
[14] McLeish DL (2005) Monte Carlo methods in finance. Wiley Finance
[15] Melnikov A, Romaniuk Y (2006) Evaluating the performance of Gompertz, Makeham and Lee-Carter mortality models for risk management. Insur Math Econ 39:310–329 · Zbl 1151.91577
[16] Pitacco E, Denuit M, Haberman S, Olivieri A (2009) Modelling longevity dynamics for pensions and annuity business. Oxford University Press · Zbl 1163.91005
[17] Renshaw AE, Haberman S (2003a) Lee-Carter mortality forecasting: a parallel generalised linear modelling approach for England and Wales mortality projections. Appl Stat 52:119–137 · Zbl 1111.62359
[18] Renshaw AE, Haberman S (2003b) On the forecasting of mortality reduction factors. Insur Math Econ 32:379–401 · Zbl 1025.62041
[19] Renshaw AE, Haberman S (2003c) Lee-Carter mortality forecasting with age specific enhancement. Insur Math Econ 33:255–272 · Zbl 1103.91371
[20] Renshaw AE, Haberman S (2008) On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee–Carter modelling. Insur Math Econ 42:797–816 · Zbl 1152.91598
[21] Ross S (2002) Simulation. Hartcourt Academic Press
[22] Willets RC (2004) The cohort effect: insights and explanations. Br Actuar J 10(4)
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