Small population bias and sampling effects in stochastic mortality modelling. (English) Zbl 1394.91201

Summary: We propose the use of parametric bootstrap methods to investigate the finite sample distribution of the maximum likelihood estimator for the parameter vector of a stochastic mortality model. Particular emphasis is placed on the effect that the size of the underlying population has on the distribution of the MLE in finite samples, and on the dependency structure of the resulting estimator: that is, the dependencies between estimators for the age, period and cohort effects in our model. In addition, we study the distribution of a likelihood ratio test statistic where we test a null hypothesis about the true parameters in our model. Finally, we apply the LRT to the cohort effects estimated from observed mortality rates for females in England and Wales and males in Scotland.


91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
91D20 Mathematical geography and demography


Full Text: DOI Link


[1] Andreev K (2002) Evolution of the Danish Population from 1835 to 2000, Vol. 9, University Press of Southern Denmark
[2] Anisimova, M; Bielawski, JP; Yang, Z, Accuracy and power of the likelihood ratio test in detecting adaptive molecular evolution, Mol Biol Evolut, 18, 1585-1592, (2001)
[3] Booth H, Hyndman RJ, Tickle L, De Jong P et al. (2006) Lee-Carter mortality forecasting: a multi-country comparison of variants and extensions, Technical report, Monash University, Department of Econometrics and Business Statistics
[4] Brouhns, N; Denuit, M; Keilegom, I, Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scand Actuar J, 3, 212-224, (2005) · Zbl 1092.91038
[5] Brouhns, N; Denuit, M; Vermunt, JK, A Poisson log-bilinear regression approach to the construction of projected lifetables, Insur Math Econ, 31, 373-393, (2002) · Zbl 1074.62524
[6] Cairns, AJG; Blake, D; Dowd, K, A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, J Risk Insur, 73, 687-718, (2006)
[7] Cairns, AJG; Blake, D; Dowd, K; Coughlan, GD; Epstein, D; Khalaf-Allah, M, Mortality density forecasts: an analysis of six stochastic mortality models, Insur Math Econ, 48, 355-367, (2011) · Zbl 1231.91179
[8] Cairns, AJG; Blake, D; Dowd, K; Coughlan, GD; Epstein, D; Ong, A; Balevich, I, A quantitative comparison of stochastic mortality models using data from england and wales and the united states, N Am Actuar J, 13, 1-35, (2009)
[9] Cairns, AJG; Blake, D; Dowd, K; Coughlan, GD; Khalaf-Allah, M, Bayesian stochastic mortality modelling for two populations, ASTIN Bull, 41, 29-59, (2011) · Zbl 1228.91032
[10] Cox DR, Hinkley DV (1979) Theoretical statistics, CRC Press
[11] Currie, ID, On Fitting generalized linear and non-linear models of mortality, Scand Actuar J, 4, 356-383, (2016) · Zbl 1401.91123
[12] Czado, C; Delwarde, A; Denuit, M, Bayesian Poisson log-bilinear mortality projections, Insur Math Econ, 36, 260-284, (2005) · Zbl 1110.62142
[13] D’Amato V, Haberman S, Russolillo M (2009) Efficient bootstrap applied to the Poisson log-bilinear Lee Carter model. Proceedings of the Applied Stochastic Models and Data Analysis (ASMDAG09), pp. 374-377
[14] Davison AC (1997) Bootstrap methods and their application, Vol. 1, Cambridge university press
[15] Ellis PD (2010) The essential guide to effect sizes: statistical power, meta-analysis, and the interpretation of research results, Cambridge University Press
[16] Gelman A, Carlin JB, Stern HS, Rubin DB (1995) Bayesian data analysis. Chapman and Hall, London · Zbl 0914.62018
[17] Huelsenbeck JP, Crandall KA (1997) Phylogeny estimation and hypothesis testing using maximum likelihood. Annual Review of Ecology and Systematics, pp 437-466
[18] Jarner, SF; Kryger, EM, Modelling adult mortality in small populations: the SAINT model, ASTIN Bull, 41, 377-418, (2011) · Zbl 1239.91128
[19] Kendall M, Stuart A, Ord J (1987) Kendalls advanced theory of statistics. Oxford University Press · Zbl 0621.62001
[20] Kleinow T, Richards SJ (2016) Parameter risk in time-series mortality forecasts, Working paper, Heriot-Watt University
[21] Kogure A, Kitsukawa K, Kurachi Y (2009) A Bayesian comparison of models for changing mortalities toward evaluating longevity risk in Japan. Asia-Pacific Journal of Risk and Insurance 3(2)
[22] Kogure, A; Kurachi, Y, A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions, Insur Math Econ, 46, 162-172, (2010) · Zbl 1231.91438
[23] Lee, RD; Carter, LR, Modeling and forecasting US mortality, Journal of the American statistical association, 87, 659-671, (1992) · Zbl 1351.62186
[24] Liu X, Braun WJ (2011) Investigating mortality uncertainty using the block bootstrap. Journal of Probability and Statistics 2010
[25] Mood A, Graybill F, Boes D (1963) Introduction into the theory of statistics · Zbl 0277.62002
[26] Neyman J, Pearson ES (1992) On the problem of the most efficient tests of statistical hypotheses, Springer
[27] Nielsen B, Nielsen JP (2014) Identification and forecasting in mortality models. The Scientific World Journal
[28] OBrien RM (2014) Estimable functions in age-period-cohort models: a unified approach. Quality & Quantity 48(1):457-474
[29] Pedroza, C, A Bayesian forecasting model: predicting US male mortality, Biostatistics, 7, 530-550, (2006) · Zbl 1170.62397
[30] Reichmuth W, Sarferaz S (2008) Bayesian Demographic Modeling and Forecasting: An Application to U.S. Mortality, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät
[31] Renshaw, AE; Haberman, S, Lee-Carter mortality forecasting with age-specific enhancement, Insur Math Econ, 33, 255-272, (2003) · Zbl 1103.91371
[32] Searle SR (1971) Linear models. New york: Wiley & Sons
[33] Wilks, SS, The large-sample distribution of the likelihood ratio for testing composite hypotheses, Ann Math Stat, 9, 60-62, (1938) · Zbl 0018.32003
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.