×

zbMATH — the first resource for mathematics

Exploring the longevity risk using statistical tools derived from the Shiryaev-Roberts procedure. (English) Zbl 1416.91144
Summary: Longevity and mortality risks are daily issues for the actuarial community. To monitor this risk, various, accurate and efficient tools have been developed (e.g., [A. N. Shiryaev, Theory Probab. Appl. 8, 22–46 (1963; Zbl 0213.43804); translation from Teor. Veroyatn. Primen. 8, 26–51 (1963); S. W. Roberts, “A comparison of some control chart procedures”, Technometrics 8, No. 3, 411–430 (1966; doi:10.1080/00401706.1966.10490374); A. S. Polunchenko and A. G. Tartakovsky, Ann. Stat. 38, No. 6, 3445–3457 (2010; Zbl 1204.62141)]). A particular attention is usually spent on the detection of a change-point, i.e., the time where the level of mortality changes [N. El Karoui et al., Ann. Appl. Probab. 27, No. 4, 2515–2538 (2017; Zbl 1378.62044); J.-C. Croix et al., “Mortality: a statistical approach to detect model misspecification”, Bull. Français d’Actuariat 15, No. 29, 75–112 (2015), https://hal.archives-ouvertes.fr/hal-00839339/]. A common assumption in all these works is that the distribution of the deaths is well known not only before the change-point but also after. In the present paper, we consider a parametric framework for the distribution after the changer and we suppose that we do not know its parameter after the change-point. Thus we focus on its estimation. Our method is derived from the sequential Shiryaev-Roberts procedure. The paper starts with a presentation of this procedure and our methodology in a general framework. We provide a specific Poisson model, designed here for the study of the mortality as in [T. E. Rhodes and S. A. Freitas, “Advanced statistical analysis of mortality”, AFIR papers. Boston Colloquia. MIB inc., Westwood Google Scholar (2004); J. Tomas and F. Planchet, Insur. Math. Econ. 63, 169–190 (2015; Zbl 1348.91184)]. Two versions are analysed: in the first one, we assume that the current mortality is stable and we look for a sudden but persistent change of level. In the second model, we introduce a new set-up: the mortality evolves at a steady pace, and we look for a change of the trend. Variants of these approaches are also widely expressed and are compared to benchmark methodologies. An important part of this work is devoted to the application of our methodology on real data, in a context where the change is obvious, using specific methodologies to adjust the data as in [Y. Mei et al., Stat. Sin. 21, No. 2, 597–624 (2011; Zbl 1214.62017)]. We also study a real insurance portfolio where no specific information might help us to understand the change, and where the change itself does not seem perceptible. For the given examples, the main results allow us to identify the change-points of the mortality when they happen and to measure the lag before clear identification of the phenomena.
MSC:
91B30 Risk theory, insurance (MSC2010)
Software:
R; snowfall
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Caselli, G; Vallin, J; Vaupel, JW; Yashin, A, Age-specific mortality trends in France and Italy Since 1900: period and cohort effects, Eur J Popul Revue européenne de démographie, 3, 33-60, (1987)
[2] Croix, J; Planchet, F; Therond, P, Mortality: a statistical approach to detect model misspecification, Bulletin Français d’Actuariat, 15, 75-112, (2015)
[3] Csörgő M, Horváth L (1997) Limit theorems in change-point analysis. Wiley series in probability and statistics. Wiley, Chichester
[4] Cutler D, Meara E (2001) Changes in the age distribution of mortality over the 20th century. Technical report, National Bureau of Economic Research
[5] Jong, P; Boyle, PP, Monitoring mortality: a state-space approach, J Econom, 23, 131-146, (1983)
[6] Didou, M. (2011). Amélioration de la mortalité : tendances passées et projections en france, en espagne, au royaume-uni et aux etats-unis
[7] El Karoui N, Loisel S, Salhi Y (2015) Minimax optimality in robust detection of a disorder time in Poisson rate. https://hal.archives-ouvertes.fr/hal-01149749 · Zbl 1378.62044
[8] Foster, DP; George, EI, Estimation up to a change-point, Ann Stat, 21, 625-644, (1993) · Zbl 0779.62018
[9] Frick, K; Munk, A; Sieling, H, Multiscale change point inference, J R Stat Soc Ser Stat Methodol, 76, 495-580, (2014) · Zbl 1411.62065
[10] Gandy, A; Jensen, U; Lütkebohmert, C, A Cox model with a change-point applied to an actuarial problem, Braz J Probab Stat, 19, 93-109, (2005) · Zbl 1272.62067
[11] Karr A (1993) Probability. Springer texts in statistics, Springer, New York
[12] Knaus, J; Porzelius, C; Binder, H; Schwarzer, G, Easier parallel computing in R with snowfall and sfcluster, R J, 1, 54-59, (2009)
[13] Lopez O, Do Huu D (2011) Étude de stabilité et élaboration de procédures de détection de rupture dans le cadre de la modélisation de sinistres automobiles
[14] Matthews, D; Farewell, V; Pyke, R; etal., Asymptotic score-statistic processes and tests for constant hazard against a change-point alternative, Ann Stat, 13, 583-591, (1985) · Zbl 0576.62032
[15] Mei, Y; Han, SW; Tsui, K-L, Early detection of a change in Poisson rate after accounting for population size effects, Stat Sin, 21, 597-624, (2011) · Zbl 1214.62017
[16] Mouyopa Djitta L (2015) Etude de l’aggravation du risque dépendance dans un contexte de réassurance
[17] Olivieri A, Pitacco E et al (2002) Inference about mortality improvements in life annuity portfolios. In: 27th International congress of actuaries, Cancun, Mexico
[18] Planchet, F; Tomas, J, Constructing entity specific mortality table: adjustment to a reference, Eur Actuar J, 4, 247-279, (2014) · Zbl 1329.91078
[19] Planchet F, Tomas J (2014) Construction et validation des références de mortalité de place. Institut des Actuaires, Note de travail, II(1291-11,v1.4)
[20] Planchet, F; Tomas, J, Prospective mortality tables: taking heterogeneity into account, Insur Math Econ, 63, 169-190, (2015) · Zbl 1348.91184
[21] Pollack, M; Tartakovsky, AG, Optimality properties of the shiryaev-Roberts procedure, Stat Sin, 19, 1729-1739, (2009) · Zbl 05629283
[22] Pollak M (2009) The Shiryaev-Roberts change-point detection procedure in retrospect—theory and practice. In: Proceedings of the 2nd international workshop on sequential methodologies, University of Technology of Troyes, Troyes, France
[23] Polunchenko, AS; Tartakovsky, AG, On optimality of the shiryaev-Roberts procedure for detecting a change in distribution, Ann Stat, 38, 3445-3457, (2010) · Zbl 1204.62141
[24] Rhodes TE, Freitas SA (2004) Advanced statistical analysis of mortality. AFIR papers. Boston Colloquia. MIB inc., Westwood
[25] Roberts, SW, A comparison of some control chart procedures, Technometrics, 8, 411-430, (1966)
[26] Servier A (2010) Etude de la stabilité et de la fiabilité des données nécessaires á la construction de tables d’expérience
[27] Shiryaev, AN, On optimum methods in quickest detection problems, Theory Probab Appl, 8, 22-46, (1963) · Zbl 0213.43804
[28] Fotopoulos, SB; Jandhyala, VK; Khapalova, E, Exact asymptotic distribution of change-point mle for change in the mean of Gaussian sequences, Ann Appl Stat, 4, 1081-1104, (2010) · Zbl 1194.62016
[29] Vallin J, Meslé F (2010) Espérance de vie : peut-on gagner trois mois par an indéfiniment ? Population et Sociétés (473)
[30] Wu Y (2007) Inference for change point and post change means after a CUSUM test, vol 180. Springer Science & Business Media, Berlin
[31] Wu, Y, Inference after truncated one-sided sequential test, Commun Stat Theory Methods, 45, 3076-3094, (2016) · Zbl 1369.62204
[32] Wu, Y, Inference for post-change parameters after sequential CUSUM test under ar(1) model, J Stat Plan Inference, 168, 52-67, (2016) · Zbl 1353.62090
[33] Zucchini W, MacDonald I (2009) Hidden markov models for time series: an introduction using R. Monographs on statistics & applied probability. CRC Press, Chapman & Hall/CRC, Boca Raton · Zbl 1180.62130
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.