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Modeling trend processes in parametric mortality models. (English) Zbl 1400.91241

Summary: Parametric mortality models like those of R. D. Lee and L. R. Carter [J. Am. Stat. Assoc. 87, No. 419, 659–675 (1992; Zbl 1351.62186)], A. J. Cairns et al. [“A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration”,J. Risk Insurance 73, No. 4, 687–718 (2006; doi:10.1111/j.1539-6975.2006.00195.x)], or R. Plat [Insur. Math. Econ. 45, No. 3, 393–404 (2009; Zbl 1231.91227)] typically include one or more time dependent parameters. Often, a random walk with drift is used to project these parameters into the future. However, longer time series of historical mortality data often show patterns which a random walk with drift is highly unlikely to generate. In fact, historical mortality trends often appear to be trend stationary around piecewise linear trends with changing slopes over time (see, e.g., [P. Sweeting, “A trend-change extension of the Cairns-Blake-Dowd model”, Ann. Actuar. Sci. 5, No. 2, 143–162 (2011; doi:10.1017/S1748499511000017)] or [J. S.-H. Li et al., “Structural changes in the Lee-Carter mortality indexes: detection and implications”, N. Am. Actuar. J. 15, No. 1, 13–31 (2011; doi:10.1080/10920277.2011.10597607)]). Periods of lower (but rather constant) mortality improvements are followed by periods of higher improvements and vice versa.
In this paper, we propose an alternative trend process which builds on the patterns observed in the historical data. Future trend changes occur randomly over time, and also the trend change magnitude is stochastic. Furthermore, we show how the parameters of this trend process, in particular the probability of observing a trend change in a certain year and the distribution for the trend change magnitude, can be estimated from historical data. We also outline how uncertainty in the parameter estimates can be accounted for. Finally, we compare the trend process to other trend processes which have been proposed in the literature.

MSC:

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

Software:

Human Mortality
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