Abstract
The Lee—Carter model is a useful dynamic stochastic model to represent the evolution of central mortality rates throughout time. This model only considers the uncertainty about the coefficient related to the mortality trend over time but not to the age-dependent coefficients. This paper proposes a fuzzy-random extension of the Lee—Carter model that allows quantifying the uncertainty of both kinds of parameters. As it is commonplace in actuarial literature, the variability of the time-dependent index is modeled as an ARIMA time series. Likewise, the uncertainty of the age-dependent coefficients is also quantified, but by using triangular fuzzy numbers. The consideration of this last hypothesis requires developing and solving a fuzzy regression model. Once the fuzzy-random extension has been introduced, it is also shown how to obtain some variables linked with central mortality rates such as death probabilities or life expectancies by using fuzzy numbers arithmetic. It is simultaneously shown the applicability of our developments with data of Spanish male population in the period 1970–2012. Finally we make a comparative assessment of our method with alternative Lee—Carter model estimates on 16 Western Europe populations.
Highlights
Classical actuarial methods graduate mortality by only taking into account the age of persons without calendar year considerations
This paper quantifies uncertain quantities as a common type of Fuzzy Number (FN), Triangular Fuzzy Numbers (TFNs), that will be symbolized as A A, l, r being A the core of the triangular FN (TFN) (μ A 1) and l and r its left and right spreads, respectively
Notes: (1) “Wins/Losses” stands for the number of cases in which basic LC (BLC) and fuzzy-random extension of the LC model (FRLC) point predictions are better/worse than FKSLC. (2) W stands for the value of the Wilcoxon rank test statistic. (3) “*”, “**” and “***” stand for the rejection of the null hypothesis with a significance level of
Summary
Classical actuarial methods graduate mortality by only taking into account the age of persons without calendar year considerations. In the last decades of the 20th century, several papers developed dynamic stochastic approaches for the evolution of mortality rates throughout calendar time and, so, projecting mortality to the future with these models became more accurate. In this way, the method in [26], that we will name LC, is one of the most extended methodologies. We finish the work by pointing out the main conclusions and suggesting possible extensions
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