# Exploring the longevity risk using statistical tools derived from the Shiryaev–Roberts procedure

## Abstract

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. Shiryaev in Theory Probab Appl 8(1):22–46, 1963; Roberts in Technometrics 8(3):411–430, 1966; Polunchenko and Tartakovsky in Ann Stat 38(6):3445–3457, 2010). A particular attention is usually spent on the detection of a change-point, i.e. the time where the level of mortality changes (El Karoui et al. in Minimax optimality in robust detection of a disorder time in poisson rate (working paper or preprint), 2015; Croix et al. in Bulletin Français d’Actuariat 15(29):75–112, 2015). 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 Rhodes and Freitas (Advanced statistical analysis of mortality. AFIR papers. Boston Colloquia. MIB inc., Westwood, 2004) and Planchet and Tomas (Insur Math Econ 63:169–190, 2015). 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 Mei et al. (Stat Sin 597–624, 2011). 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.

## Keywords

Shiryaev–Roberts process Change-point Discrete Poisson model Sequential estimation Longevity risk Mortality trend Mortality jumps## References

- 1.Caselli G, Vallin J, Vaupel JW, Yashin A (1987) Age-specific mortality trends in france and italy since 1900: period and cohort effects. Eur J Popul Revue européenne de démographie 3(1):33–60CrossRefGoogle Scholar
- 2.Croix J, Planchet F, Therond P (2015) Mortality: a statistical approach to detect model misspecification. Bulletin Français d’Actuariat 15(29):75–112Google Scholar
- 3.Csörgő M, Horváth L (1997) Limit theorems in change-point analysis. Wiley series in probability and statistics. Wiley, ChichesterGoogle Scholar
- 4.Cutler D, Meara E (2001) Changes in the age distribution of mortality over the 20th century. Technical report, National Bureau of Economic ResearchGoogle Scholar
- 5.De Jong P, Boyle PP (1983) Monitoring mortality: a state-space approach. J Econom 23(1):131–146CrossRefGoogle Scholar
- 6.Didou, M. (2011). Amélioration de la mortalité : tendances passées et projections en france, en espagne, au royaume-uni et aux etats-unisGoogle Scholar
- 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
- 8.Foster DP, George EI (1993) Estimation up to a change-point. Ann Stat 21(2):625–644MathSciNetCrossRefMATHGoogle Scholar
- 9.Frick K, Munk A, Sieling H (2014) Multiscale change point inference. J R Stat Soc Ser Stat Methodol 76(3):495–580MathSciNetCrossRefGoogle Scholar
- 10.Gandy A, Jensen U, Lütkebohmert C (2005) A Cox model with a change-point applied to an actuarial problem. Braz J Probab Stat 19(2):93–109MathSciNetMATHGoogle Scholar
- 11.Karr A (1993) Probability. Springer texts in statistics, Springer, New YorkGoogle Scholar
- 12.Knaus J, Porzelius C, Binder H, Schwarzer G (2009) Easier parallel computing in R with snowfall and sfCluster. R J 1(1):54–59Google Scholar
- 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 automobilesGoogle Scholar
- 14.Matthews D, Farewell V, Pyke R et al (1985) Asymptotic score-statistic processes and tests for constant hazard against a change-point alternative. Ann Stat 13(2):583–591MathSciNetCrossRefMATHGoogle Scholar
- 15.Mei Y, Han SW, Tsui K-L (2011) Early detection of a change in Poisson rate after accounting for population size effects. Stat Sin 21(2):597–624MathSciNetCrossRefMATHGoogle Scholar
- 16.Mouyopa Djitta L (2015) Etude de l’aggravation du risque dépendance dans un contexte de réassuranceGoogle Scholar
- 17.Olivieri A, Pitacco E et al (2002) Inference about mortality improvements in life annuity portfolios. In: 27th International congress of actuaries, Cancun, MexicoGoogle Scholar
- 18.Planchet F, Tomas J (2014) Constructing entity specific mortality table: adjustment to a reference. Eur Actuar J 4(2):247–279MathSciNetCrossRefMATHGoogle Scholar
- 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)Google Scholar
- 20.Planchet F, Tomas J (2015) Prospective mortality tables: taking heterogeneity into account. Insur Math Econ 63:169–190MathSciNetCrossRefMATHGoogle Scholar
- 21.Pollack M, Tartakovsky AG (2009) Optimality properties of the Shiryaev–Roberts procedure. Stat Sin 19:1729–1739MathSciNetMATHGoogle Scholar
- 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, FranceGoogle Scholar
- 23.Polunchenko AS, Tartakovsky AG (2010) On optimality of the Shiryaev–Roberts procedure for detecting a change in distribution. Ann Stat 38(6):3445–3457MathSciNetCrossRefMATHGoogle Scholar
- 24.Rhodes TE, Freitas SA (2004) Advanced statistical analysis of mortality. AFIR papers. Boston Colloquia. MIB inc., WestwoodGoogle Scholar
- 25.Roberts SW (1966) A comparison of some control chart procedures. Technometrics 8(3):411–430MathSciNetCrossRefGoogle Scholar
- 26.Servier A (2010) Etude de la stabilité et de la fiabilité des données nécessaires á la construction de tables d’expérienceGoogle Scholar
- 27.Shiryaev AN (1963) On optimum methods in quickest detection problems. Theory Probab Appl 8(1):22–46MathSciNetCrossRefMATHGoogle Scholar
- 28.Fotopoulos SB, Jandhyala VK, Khapalova E (2010) Exact asymptotic distribution of change-point mle for change in the mean of gaussian sequences. Ann Appl Stat 4(2):1081–1104MathSciNetCrossRefMATHGoogle Scholar
- 29.Vallin J, Meslé F (2010) Espérance de vie : peut-on gagner trois mois par an indéfiniment ?
*Population et Sociétés*(473)Google Scholar - 30.Wu Y (2007) Inference for change point and post change means after a CUSUM test, vol 180. Springer Science & Business Media, BerlinGoogle Scholar
- 31.Wu Y (2016a) Inference after truncated one-sided sequential test. Commun Stat Theory Methods 45(10):3076–3094MathSciNetCrossRefMATHGoogle Scholar
- 32.Wu Y (2016b) Inference for post-change parameters after sequential CUSUM test under ar(1) model. J Stat Plan Inference 168:52–67MathSciNetCrossRefMATHGoogle Scholar
- 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 RatonGoogle Scholar