×

Extreme value analysis of mortality at the oldest ages: a case study based on individual ages at death. (English) Zbl 1414.91190

Summary: In this article, the force of mortality at the oldest ages is studied using the statistical tools from extreme value theory. A unique data basis recording all individual ages at death above 95 for extinct cohorts born in Belgium between 1886 and 1904 is used to illustrate the relevance of the proposed approach. No leveling off in the force of mortality at the oldest ages is found, and the analysis supports the existence of an upper limit to human lifetime for these cohorts. Therefore, assuming that the force of mortality becomes ultimately constant, that is, that the remaining lifetime tends to the negative exponential distribution as the attained age grows is a conservative strategy for managing life annuities.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
60G70 Extreme value theory; extremal stochastic processes
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Aarssen, K.; de Haan, L., On the Maximal Life Span of Humans, Mathematical Population Studies, 4, 259-281, (1994) · Zbl 0872.62096
[2] Balkema, A.; de Haan, L., Residual Life Time at Great Age, Annals of Probability, 2, 792-804, (1974) · Zbl 0295.60014
[3] Beirlant, J.; Dierckx, G.; Guillou, A., Estimation of the Extreme-Value Index and Generalized Quantile Plots, Bernoulli, 11, 949-970, (2005) · Zbl 1123.62034
[4] Beirlant, J.; Goegebeur, J.; Teugels, J.; Segers, J.; Waal, D. D.; Ferro, C., Statistics of Extremes: Theory and Applications, (2005), New York: Wiley, New York
[5] Bravo, J.; Coelho, E.; Magalhaes, M., Mortality and Longevity Projections for the Oldest-Old in Portugal, Proceedings of the European Population Conference, 1-16, (2008), Barcelona, Spain
[6] Bravo, J.; Real, P., Modeling Longevity Risk using Extreme Value Theory: An Empirical Investigation Using Portuguese and Spanish Population Data, (2012)
[7] Brillinger, D. R., A Justification of Some Common Laws of Mortality, Transactions of the Society of Actuaries, 13, 116-119, (1961)
[8] Cebrian, A. C.; Denuit, M.; Lambert, P., Generalized Pareto Fit to the Society of Actuaries’ Large Claims Database, North American Actuarial Journal, 7, 18-36, (2003) · Zbl 1084.62108
[9] Dekkers, A.; Einmahl, J.; de Haan, L., A Moment Estimator for the Index of an Extreme-Value Distribution, Annals of Statistics, 17, 1833-1855, (1989) · Zbl 0701.62029
[10] Dong, X.; Milholland, B.; Vijg, J., Evidence for a Limit to Human Lifespan, Nature, 538, 257-259, (2016)
[11] Einmahl, J. H.; Magnus, J. R., Records in Athletics through Extreme-Value Theory, Journal of the American Statistical Association, 103, 1382-1391, (2008) · Zbl 1286.62047
[12] Einmahl, J. H.; Smeets, S. G., Ultimate 100-M World Records through Extreme-Value Theory, Statistica Neerlandica, 65, 32-42, (2011)
[13] Fraga Alves, I.; Neves, C.; Rosario, P., A General Estimator for the Right Endpoint with an Application to Supercentenarian Women’s Records, Extremes, 20, 1, 199-237, (2017) · Zbl 1367.62160
[14] Gampe, J., Human Mortality beyond Age 110, In Supercentenarians, 219-230, (2010), Berlin: Springer, Berlin
[15] Gavrilov, L. A.; Gavrilova, N. S., Mortality Measurement at Advanced Ages: A Study of the Social Security Administration Death Master File, North American Actuarial Journal, 15, 432-447, (2011)
[16] Gumbel, E. J., La Durée extrême de la vie humaine, (1937), Paris: Hermann, Paris
[17] Han, Z., Living to 100 and Beyond: An Extreme Value Study, (2005)
[18] Hanayama, N.; Sibuya, M., Estimating the Upper Limit of Lifetime Probability Distribution, Based on Data of Japanese Centenarians, Journals of Gerontology: Biological Sciences, 71, 8, 1014-1021, (2016)
[19] Hosking, J.; Wallis, J., Parameter and Quantile Estimation for the Generalized Pareto Distribution, Technometrics, 29, 339-349, (1987) · Zbl 0628.62019
[20] Kannisto, V., Development of Oldest-Old Mortality 1950–1990: Evidence from 28 Developed Countries, Odense Monographs on Population Aging, (1994)
[21] Kiefer, J., K-Sample Analogues of the Kolmogorov-Smirnov and Cramer-Von Mises Tests, Annals of Mathematical Statistics, 30, 420-447, (1956) · Zbl 0134.36707
[22] Li, J.; Ng, A.; Chan, W., Modeling Old-Age Mortality Risk for the Populations of Australia and New Zealand: An Extreme Value Approach, Mathematics and Computers in Simulation, 81, 1325-1333, (2010) · Zbl 1217.62177
[23] Li, J. S. H.; Hardy, M. R.; Tan, K. S., Threshold Life Tables and Their Applications, North American Actuarial Journal, 12, 99-115, (2008)
[24] MacDonald, A.; Scarrott, C. J.; Lee, D.; Darlow, B.; Reale, M.; Russell, G., A Flexible Extreme Value Mixture Model, Computational Statistics and Data Analysis, 55, 2137-2157, (2011) · Zbl 1328.62296
[25] Manton, K. G.; Stallard, E.; Vaupel, J. W., Alternative Models for the Heterogeneity of Mortality Risks among the Aged, Journal of the American Statistical Association, 81, 635-644, (1986)
[26] Neves, C.; Alves, M. I. F., Reiss and Thomas’ Automatic Selection of the Number of Extremes, Computational Statistics and Data Analysis, 47, 689-704, (2004) · Zbl 1430.62096
[27] Ouellette, N.; Bourbeau, R., The Trajectory of Mortality at Ages 100 and Beyond: An Analysis of Individual Level Data in Canada, (2014)
[28] Pickands, J., Statistical Inference Using Extreme Order Statistics, Annals of Statistics, 3, 119-131, (1975) · Zbl 0312.62038
[29] Poulain, M., Le registre de population belge, Histoire de la population de la Belgique et de ses territoires, 83-116, (2010)
[30] Poulain, M.; Chambre, D.; Foulon, M., Survival among Belgian Centenarians (1870–1894 Cohorts), Population, 13, 117-138, (2001)
[31] Reiss, R.-D.; Thomas, M., Statistical Analysis of Extreme Values, with Applications to Insurance, Finance, Hydrology and Other Fields, (1997) · Zbl 0880.62002
[32] Robine, J. M.; Cournil, A.; Gampe, J.; Vaupel, J. W., IDL, the International Database on Longevity, (2005)
[33] Scarrott, C.; MacDonald, A., A Review of Extreme Value threshold Estimation and Uncertainty Quantification, REVSTAT-Statistical Journal, 10, 33-60, (2012) · Zbl 1297.62120
[34] Wang, J.-L.; Muller, H.-G.; Capra, B. W., Analysis of Old-Oldest Mortality: Lifetables Revisited, Annals of Statistics, 26, 126-163, (1998) · Zbl 0930.62040
[35] Watts, K.; Dupuis, D.; Jones, B., An Extreme Value Analysis of Advance Age Mortality Data, North American Actuarial Journal, 10, 162-178, (2006)
[36] Wilmoth, J. R.; Robine, J. M., The World Trend in Maximum Life Span, Population and Development Review, 29, 239-257, (2003)
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.