高龄人口死亡率预测模型的比较与选择
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摘要
高龄人口死亡率预测模型是人口预测、养老金成本和债务评估以及长寿风险度量与管理的基础。我国大陆地区高龄人口死亡数据量少、数据波动性大,如何选择适合我国高龄数据特点的死亡率预测模型,是重要的研究课题。本文在归纳总结死亡率预测模型研究进展的基础上,先采用数据较为充分的台湾地区高龄死亡数据,选用Lee-Carter、CBD、贝叶斯分层模型等八种死亡率模型,对模型的拟合效果、预测效果和稳健性做出比较。在此基础上,基于修正和平滑后的我国大陆人口死亡数据,采用CBD模型和贝叶斯分层模型建模和预测。结果显示:贝叶斯分层模型能捕捉我国大陆高龄死亡率数据的历史波动,预测区间能够涵盖全部死亡率的真实值,但预测区间过宽,生存曲线不收敛;相比之下,CBD模型对我国大陆地区高龄死亡率的拟合和预测较好,预测区间和生存曲线合理。在长寿风险度量中,建议采用CBD模型。
The mortality modeling of senior population is the basis for population forecasting,pension cost and debt assessment,and longevity risk management.In mainland China,age-specific mortality data of the elderly population is very limited and volatile.How to choose a suitable mortality model for China's senior age data has become an important research topic.This paper reviewed researches on mortality prediction models,and the senior population mortality data from Taiwan was adopted to fit the eight mortality models commonly used in the literature such as LC,CBD and Bayesian hierarchical model and compared the fitting and predictive effects and the robustness.Next,the paper modeled and projected the smoothed senior age mortality data in mainland China with CBD and Bayesian hierarchical model.The results showed the Bayesian hierarchical model could capture the historical fluctuations of the data and the prediction intervals covered the true value of mortality but had a broader interval and unreasonable survival curve.In comparison,the CBD model had better fitting and forecasting effects,with reasonable prediction intervals and survival curve.Therefore,for the longevity risk measurement,CBD mortality model was recommended.
The mortality modeling of senior population is the basis for population forecasting,pension cost and debt assessment,and longevity risk management.In mainland China,age-specific mortality data of the elderly population is very limited and volatile.How to choose a suitable mortality model for China's senior age data has become an important research topic.This paper reviewed researches on mortality prediction models,and the senior population mortality data from Taiwan was adopted to fit the eight mortality models commonly used in the literature such as LC,CBD and Bayesian hierarchical model and compared the fitting and predictive effects and the robustness.Next,the paper modeled and projected the smoothed senior age mortality data in mainland China with CBD and Bayesian hierarchical model.The results showed the Bayesian hierarchical model could capture the historical fluctuations of the data and the prediction intervals covered the true value of mortality but had a broader interval and unreasonable survival curve.In comparison,the CBD model had better fitting and forecasting effects,with reasonable prediction intervals and survival curve.Therefore,for the longevity risk measurement,CBD mortality model was recommended.
引文
[1] 段白鸽.我国全年龄段人口平均预期寿命的动态演变[J].人口与经济,2015,(1):49-63.
[2] 李志生,刘恒甲.Lee-Carter死亡率模型的估计与应用——基于中国人口数据的分析[J].中国人口科学,2010,(3):46-56.
[3] 王晓军,黄顺林.中国人口死亡率随机预测模型的比较与选择[J].人口与经济,2011,(1):82-86.
[4] 王晓军,路倩.我国商业养老年金的供需困境探讨:基于年金价值和长寿风险的视角[J].保险研究,2018,(9):13-21.
[5] 吴晓坤,王晓军.中国人口死亡率Lee-Carter模型的再抽样估计、预测与应用[J].中国人口科学,2014,(4):27-34.
[6] 祝伟,陈秉正.中国城市人口死亡率的预测[J].数理统计与管理,2009,28(4):736-744.
[7] Abel G,Bijak J,Forster J,et al.What Do Bayesian Methods Offer Population Forecasters?[R].Technical report,ESRC Centre for Population Change,2010.
[8] Alexander M,Zagheni E,Barbieri M .A Flexible Bayesian Model for Estimating Subnational Mortality[J].Demography,2017,54(2):1-17.
[9] Alkema L,Raftery A,Gerland P,et al.Estimating the Total Fertility Rate from Multiple Imperfect Data Sources and Assessing Its Un-certainty[R].Technical Report,Centre for Statistics and the Social Sciences,University of Washington.Working paper 89,2008.
[10] Alkema L,Raftery A,Gerland P,et al.Probabilistic Projections of the Total Fertility Rate for All Countries[J].Demography,2011,43(3):815-839.
[11] Alkema L,Raftery A,Gerland P,et al.“Estimating Trends in the Total Fertility Rate with Uncertainty Using Imperfect Data:Examples from West Africa[J].Demographic Research,2012.26:331-362.
[12] Berkum F V,Antonio K,Vellekoop M.The Impact of Multiple Structural Changes on Mortality Predictions[J].Social Science Electronic Publishing,2014,2016(7):1-22.
[13] Bijak J,Wi?niowski A.Bayesian Forecasting of Immigration to Selected European Countries by Using Expert Knowledge[J].Journal of the Royal Statistical Society:Series A(Statistics in Society),2010,173(4):775-796.
[14] Brierley M,Forster J,McDonald J,Smith P.Bayesian Estimation of Migration Flows.In Raymer J,Willekens F.(eds.),International Migration in Europe:Data,Models and Estimates[M].Wiley and Sons,2008.
[15] B?rger M,Fleischer D,Kuksin N.Modeling the Mortality Trend under Modern Solvency Regimes[J].ASTIN Bulletin,2013,44(1),1-38.
[16] Brouhns N,Denuit M,Vermunt J K.A Poisson Log-bilinear Regression Approach to the Construction of Projected Lifetables[J].Insurance Mathematics & Economics,2002,31(3):373-393.
[17] Bryant J R,Graham P J.Bayesian Demographic Accounts:Subnational Population Estimation Using Multiple Data Sources[J].Bayesian Analysis,2013,8(3):591-622.
[18] Cairns A J G,Blake D,Dowd K.A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty:Theory and Calibration[J].Journal of Risk and Insurance.2006,73:687-718.
[19] Cairns A J G,Blake D,Dowd K,et al.A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States[J].North American Actuarial Journal,2009,1-35.
[20] Cairns A J G,Blake D,Dowd K,etal.Mortality Density Forecasts:An Analysis of Six Stochastic Mortality Models[J].Insurance Mathematics & Economics,2011a,48(3):355-367.
[21] Cairns A J G,Blake A,Dowd D,et al.Bayesian Stochastic Mortality Modelling for Two Populations[J].Astin Bulletin,2011b,41(1):29-59.
[22] Callot L,Haldrup N,Kallestrup-Lamb M.Deterministic and Stochastic Trends in the Lee-Carter Mortality Model[J].Applied Economics Letters,2016,23(7):486-493.
[23] Christiansen M C,Spodarev E,Unseld V.Differences in European Mortality Rates:A Geometric Approach on the Age-Period Plane[J].Astin Bulletin 45(3),477-502.
[24] Congdon P.Models forMigration Age Schedules:A Bayesian Perspective with An Application to Flows between Scotland and England.In Raymer J,Willekens F.(eds.),International Migration in Europe:Data,Models and Estimates[M].Wiley and Sons,2008.
[25] Congdon P.EstimatingLife Expectancies for US Small Areas:A Regression Framework[J].Journal of Geographical Systems,2014,16(1):1-18.
[26] Continuous Mortality Investigation Bureau.Standard Tables of Mortality Based on the 1979-82 Experiences[R].CMI Report No.10.London:Institute of Actuaries and Faculty of Actuaries,1990.
[27] Currie,I.D.,2006.Smoothing and Forecasting Mortality Rates with P-splines[R].Talk given at the Institute of Actuaries,June 2006.
[28] Dellaportas P,Smith A F M,Stavropoulos P .Bayesian Analysis of Mortality Data[J].Journal of the Royal Statistical Society,2001,164(2):275-291.
[29] Dowd K,Cairns A J G,Blake D,et al.Backtesting Stochastic Mortality Models:An Ex-Post Evaluation of Multi-Period Ahead-Density Forecasts[J].North American Actuarial Journal,2009,14(3):281-298.
[30] Dowd K,Cairns A J G,Blake D,et al.Evaluating the Goodness of Fit of Stochastic Mortality Models[J].Insurance Mathematics & Economics,2010,47(3):255-265.
[31] Enchev V,Kleinow T,Cairns A J G .Multi-population Mortality Models:Fitting,Forecasting and Comparisons[J].Scandinavian Actuarial Journal,2017(4):1-24.
[32] Girosi F.and King G.Demographic Forecasting[M].Princeton University Press,Princeton,2008.
[33] Haberman S,Renshaw A.On Age-Period-Cohort Parametric Mortality Rate Projections[J].Insurance:Mathematics and Economics,2009,45(2):255-270.
[34] Haberman S,Renshaw A.A Comparative Study of Parametric Mortality Projection Models[J].Insurance Mathematics and Economics,2011,48(1):35-55.
[35] Hyndman R J,Ullah M S.Robust Forecasting of Mortality and Fertility Rates:A Functional Data Approach[J].Computational Statistics and Data Analysis,2007,51(10):4942-4956.
[36] Kogure A,Kitsukawa K,Kurachi Y.A Bayesian Comparison of Models for Changing Mortalities toward Evaluating Longevity Risk in Japan[J].Asia Pacific Journal of Risk and Insurance,2009,3(2):1-21.
[37] Kogure A,Kurachi Y.A Bayesian Approach to Pricing Longevity Risk based on Risk-Neutral Predictive Distributions[J].Insurance:Mathematics and Economics,2010,46:162-172.
[38] Lee R D,Carter L R.Modeling and Forecasting U.S.Mortality[J].Journal of the American Statistical Association,1992,87,659-671.
[39] Li J.An Application of MCMCS Imulation in Mortality Projection for Populations with Limited Data[J].Demographic Research,2014,30(1):1-48.
[40] LynchS,Brown J.Obtaining Multistate Life Table Distributions for Highly Refined Subpopulations from Cross-Sectional Data:A Bayesian Extension of Sullivan’s Method[J].Demography,2010,47(4):1053-1077.
[41] Milidonis A,Efthymiou M .Mortality Leads and Lags[J].Social Science Electronic Publishing,2015,84.
[42] Mitchell D,Brockett P,Mendoza-Arriaga R,et al.Modeling and Forecasting Mortality Rates[J].Insurance Mathematics & Economics,2013,52(2):275-285.
[43] O’Hare C,Li Y.ExplainingYoung Mortality[J].Insurance:Mathematics and Economics,2012,50(1),12-25.
[44] Pedroza C.A Bayesian Forecasting Model:Predicting U.S.Male Mortality[J].Biostatistics,2006,7(4):530.
[45] Plat R.OnStochastic Mortality Modeling[J],Insurance:Mathematics and Economics.2009a,45:393-404.
[46] Plat R.Stochastic Portfolio Specific Mortality and the Quantification of Mortality Basis Risk.Insurance:Mathematics and Economics,2009b,45(1):123-132.
[47] Plat R.One-year Value-at-Risk for Longevity and Mortality[J].Insurance Mathematics & Economics,2011,49(3):462-470.
[48] Raftery A,Li N,?evcíková H,et al.Bayesian Probabilistic Population Projections for All Countries[J].Proceedings of the National Academy of Sciences,2012,109(35):13915-13921.
[49] Reichmuth W,Sarferaz S.Bayesian Demographic Modelling and Forecasting:An Application to US Mortality[R].2008,SFB 649 Discussion paper 2008-052.
[50] Rendall M,Handcock M,Jonsson S.Bayesian Estimation of Hispanic fertility Hazards from Survey and Population Data[J].Demography,2009,46(1):65-83.
[51] Renshaw A E,Haberman S.Modeling for Mortality Reduction Factors[R].Actuarial Research Paper No.127.Department of Actuarial Science and Statistics,City University,London,2000.
[52] Renshaw A E,Haberman S.On the Forecasting of Mortality Reduction Factors[J].Insurance:Mathematics and Economics,2003b,32:379-401.
[53] Renshaw A E,Haberman S.A Cohort-Based Extension to the Lee-Carter Model for Mortality Reduction Factors[J].Insurance Mathematics & Economics,2006,38(3):556-570.
[54] Yang B,Li J,Balasooriya U.Using Bootstrapping to Incorporate Model Error for Risk-Neutral Pricing of Longevity Risk[J].Insurance Mathematics and Economics,2015,62:16-27.
[55] Zhu W,Tan K S,Wang C.ModelingMulticountry Longevity Risk With Mortality Dependence:A Lévy Subordinated Hierarchical Archimedean Copulas Approach[J].Journal of Risk & Insurance,2017,84(S1):477-493.
(1)人类死亡数据库由美国加州大学伯克利分校人口系和德国罗斯托克普朗克人口研究所于2002年共同建立,网址http://www.mortality.org.
(2)由于历史的特殊性,台湾地区1925～1940年出生的队列效应较1925年前出生的队列效应在趋势上有较大改变,详见附录。1925～1940年出生的队列在2010年到达70～85岁,因而75岁、85岁的死亡率在2010左右出现波动。
(3)在贝叶斯方法中,死亡率的预测值取预测区间的中位数。
(4)对于开放区间的拆分,首先采用Kannisto模型对(x*-20)岁以上的高龄死亡率进行拟合(x*为区间下限),然后用Kannisto模型外推比x*更高年龄的死亡率,并在Lexis三角形中推导死亡人数的分布,详见Human Mortality Database,http://www.mortality.org/Public/Docs/MethodsProtocol.pdf.
(5)对于死亡人数的异常波动,首先对死亡率进行一阶差分得到差分序列ΔD(x),然后用三次样条拟合差分序列,计算差分序列的趋势f(x)和标准偏差σ,若差分序列的某一点满足:|ΔD(x) -f(x)|>1.8σ,则认为该点为离群值,并将该点调整为:ΔD(x)′ =f(x)±1.8σ。详见Human Mortality Database,http://www.mortality.org/Public/Docs/MethodsProtocol.pdf.
[2] 李志生,刘恒甲.Lee-Carter死亡率模型的估计与应用——基于中国人口数据的分析[J].中国人口科学,2010,(3):46-56.
[3] 王晓军,黄顺林.中国人口死亡率随机预测模型的比较与选择[J].人口与经济,2011,(1):82-86.
[4] 王晓军,路倩.我国商业养老年金的供需困境探讨:基于年金价值和长寿风险的视角[J].保险研究,2018,(9):13-21.
[5] 吴晓坤,王晓军.中国人口死亡率Lee-Carter模型的再抽样估计、预测与应用[J].中国人口科学,2014,(4):27-34.
[6] 祝伟,陈秉正.中国城市人口死亡率的预测[J].数理统计与管理,2009,28(4):736-744.
[7] Abel G,Bijak J,Forster J,et al.What Do Bayesian Methods Offer Population Forecasters?[R].Technical report,ESRC Centre for Population Change,2010.
[8] Alexander M,Zagheni E,Barbieri M .A Flexible Bayesian Model for Estimating Subnational Mortality[J].Demography,2017,54(2):1-17.
[9] Alkema L,Raftery A,Gerland P,et al.Estimating the Total Fertility Rate from Multiple Imperfect Data Sources and Assessing Its Un-certainty[R].Technical Report,Centre for Statistics and the Social Sciences,University of Washington.Working paper 89,2008.
[10] Alkema L,Raftery A,Gerland P,et al.Probabilistic Projections of the Total Fertility Rate for All Countries[J].Demography,2011,43(3):815-839.
[11] Alkema L,Raftery A,Gerland P,et al.“Estimating Trends in the Total Fertility Rate with Uncertainty Using Imperfect Data:Examples from West Africa[J].Demographic Research,2012.26:331-362.
[12] Berkum F V,Antonio K,Vellekoop M.The Impact of Multiple Structural Changes on Mortality Predictions[J].Social Science Electronic Publishing,2014,2016(7):1-22.
[13] Bijak J,Wi?niowski A.Bayesian Forecasting of Immigration to Selected European Countries by Using Expert Knowledge[J].Journal of the Royal Statistical Society:Series A(Statistics in Society),2010,173(4):775-796.
[14] Brierley M,Forster J,McDonald J,Smith P.Bayesian Estimation of Migration Flows.In Raymer J,Willekens F.(eds.),International Migration in Europe:Data,Models and Estimates[M].Wiley and Sons,2008.
[15] B?rger M,Fleischer D,Kuksin N.Modeling the Mortality Trend under Modern Solvency Regimes[J].ASTIN Bulletin,2013,44(1),1-38.
[16] Brouhns N,Denuit M,Vermunt J K.A Poisson Log-bilinear Regression Approach to the Construction of Projected Lifetables[J].Insurance Mathematics & Economics,2002,31(3):373-393.
[17] Bryant J R,Graham P J.Bayesian Demographic Accounts:Subnational Population Estimation Using Multiple Data Sources[J].Bayesian Analysis,2013,8(3):591-622.
[18] Cairns A J G,Blake D,Dowd K.A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty:Theory and Calibration[J].Journal of Risk and Insurance.2006,73:687-718.
[19] Cairns A J G,Blake D,Dowd K,et al.A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States[J].North American Actuarial Journal,2009,1-35.
[20] Cairns A J G,Blake D,Dowd K,etal.Mortality Density Forecasts:An Analysis of Six Stochastic Mortality Models[J].Insurance Mathematics & Economics,2011a,48(3):355-367.
[21] Cairns A J G,Blake A,Dowd D,et al.Bayesian Stochastic Mortality Modelling for Two Populations[J].Astin Bulletin,2011b,41(1):29-59.
[22] Callot L,Haldrup N,Kallestrup-Lamb M.Deterministic and Stochastic Trends in the Lee-Carter Mortality Model[J].Applied Economics Letters,2016,23(7):486-493.
[23] Christiansen M C,Spodarev E,Unseld V.Differences in European Mortality Rates:A Geometric Approach on the Age-Period Plane[J].Astin Bulletin 45(3),477-502.
[24] Congdon P.Models forMigration Age Schedules:A Bayesian Perspective with An Application to Flows between Scotland and England.In Raymer J,Willekens F.(eds.),International Migration in Europe:Data,Models and Estimates[M].Wiley and Sons,2008.
[25] Congdon P.EstimatingLife Expectancies for US Small Areas:A Regression Framework[J].Journal of Geographical Systems,2014,16(1):1-18.
[26] Continuous Mortality Investigation Bureau.Standard Tables of Mortality Based on the 1979-82 Experiences[R].CMI Report No.10.London:Institute of Actuaries and Faculty of Actuaries,1990.
[27] Currie,I.D.,2006.Smoothing and Forecasting Mortality Rates with P-splines[R].Talk given at the Institute of Actuaries,June 2006.
[28] Dellaportas P,Smith A F M,Stavropoulos P .Bayesian Analysis of Mortality Data[J].Journal of the Royal Statistical Society,2001,164(2):275-291.
[29] Dowd K,Cairns A J G,Blake D,et al.Backtesting Stochastic Mortality Models:An Ex-Post Evaluation of Multi-Period Ahead-Density Forecasts[J].North American Actuarial Journal,2009,14(3):281-298.
[30] Dowd K,Cairns A J G,Blake D,et al.Evaluating the Goodness of Fit of Stochastic Mortality Models[J].Insurance Mathematics & Economics,2010,47(3):255-265.
[31] Enchev V,Kleinow T,Cairns A J G .Multi-population Mortality Models:Fitting,Forecasting and Comparisons[J].Scandinavian Actuarial Journal,2017(4):1-24.
[32] Girosi F.and King G.Demographic Forecasting[M].Princeton University Press,Princeton,2008.
[33] Haberman S,Renshaw A.On Age-Period-Cohort Parametric Mortality Rate Projections[J].Insurance:Mathematics and Economics,2009,45(2):255-270.
[34] Haberman S,Renshaw A.A Comparative Study of Parametric Mortality Projection Models[J].Insurance Mathematics and Economics,2011,48(1):35-55.
[35] Hyndman R J,Ullah M S.Robust Forecasting of Mortality and Fertility Rates:A Functional Data Approach[J].Computational Statistics and Data Analysis,2007,51(10):4942-4956.
[36] Kogure A,Kitsukawa K,Kurachi Y.A Bayesian Comparison of Models for Changing Mortalities toward Evaluating Longevity Risk in Japan[J].Asia Pacific Journal of Risk and Insurance,2009,3(2):1-21.
[37] Kogure A,Kurachi Y.A Bayesian Approach to Pricing Longevity Risk based on Risk-Neutral Predictive Distributions[J].Insurance:Mathematics and Economics,2010,46:162-172.
[38] Lee R D,Carter L R.Modeling and Forecasting U.S.Mortality[J].Journal of the American Statistical Association,1992,87,659-671.
[39] Li J.An Application of MCMCS Imulation in Mortality Projection for Populations with Limited Data[J].Demographic Research,2014,30(1):1-48.
[40] LynchS,Brown J.Obtaining Multistate Life Table Distributions for Highly Refined Subpopulations from Cross-Sectional Data:A Bayesian Extension of Sullivan’s Method[J].Demography,2010,47(4):1053-1077.
[41] Milidonis A,Efthymiou M .Mortality Leads and Lags[J].Social Science Electronic Publishing,2015,84.
[42] Mitchell D,Brockett P,Mendoza-Arriaga R,et al.Modeling and Forecasting Mortality Rates[J].Insurance Mathematics & Economics,2013,52(2):275-285.
[43] O’Hare C,Li Y.ExplainingYoung Mortality[J].Insurance:Mathematics and Economics,2012,50(1),12-25.
[44] Pedroza C.A Bayesian Forecasting Model:Predicting U.S.Male Mortality[J].Biostatistics,2006,7(4):530.
[45] Plat R.OnStochastic Mortality Modeling[J],Insurance:Mathematics and Economics.2009a,45:393-404.
[46] Plat R.Stochastic Portfolio Specific Mortality and the Quantification of Mortality Basis Risk.Insurance:Mathematics and Economics,2009b,45(1):123-132.
[47] Plat R.One-year Value-at-Risk for Longevity and Mortality[J].Insurance Mathematics & Economics,2011,49(3):462-470.
[48] Raftery A,Li N,?evcíková H,et al.Bayesian Probabilistic Population Projections for All Countries[J].Proceedings of the National Academy of Sciences,2012,109(35):13915-13921.
[49] Reichmuth W,Sarferaz S.Bayesian Demographic Modelling and Forecasting:An Application to US Mortality[R].2008,SFB 649 Discussion paper 2008-052.
[50] Rendall M,Handcock M,Jonsson S.Bayesian Estimation of Hispanic fertility Hazards from Survey and Population Data[J].Demography,2009,46(1):65-83.
[51] Renshaw A E,Haberman S.Modeling for Mortality Reduction Factors[R].Actuarial Research Paper No.127.Department of Actuarial Science and Statistics,City University,London,2000.
[52] Renshaw A E,Haberman S.On the Forecasting of Mortality Reduction Factors[J].Insurance:Mathematics and Economics,2003b,32:379-401.
[53] Renshaw A E,Haberman S.A Cohort-Based Extension to the Lee-Carter Model for Mortality Reduction Factors[J].Insurance Mathematics & Economics,2006,38(3):556-570.
[54] Yang B,Li J,Balasooriya U.Using Bootstrapping to Incorporate Model Error for Risk-Neutral Pricing of Longevity Risk[J].Insurance Mathematics and Economics,2015,62:16-27.
[55] Zhu W,Tan K S,Wang C.ModelingMulticountry Longevity Risk With Mortality Dependence:A Lévy Subordinated Hierarchical Archimedean Copulas Approach[J].Journal of Risk & Insurance,2017,84(S1):477-493.
(1)人类死亡数据库由美国加州大学伯克利分校人口系和德国罗斯托克普朗克人口研究所于2002年共同建立,网址http://www.mortality.org.
(2)由于历史的特殊性,台湾地区1925～1940年出生的队列效应较1925年前出生的队列效应在趋势上有较大改变,详见附录。1925～1940年出生的队列在2010年到达70～85岁,因而75岁、85岁的死亡率在2010左右出现波动。
(3)在贝叶斯方法中,死亡率的预测值取预测区间的中位数。
(4)对于开放区间的拆分,首先采用Kannisto模型对(x*-20)岁以上的高龄死亡率进行拟合(x*为区间下限),然后用Kannisto模型外推比x*更高年龄的死亡率,并在Lexis三角形中推导死亡人数的分布,详见Human Mortality Database,http://www.mortality.org/Public/Docs/MethodsProtocol.pdf.
(5)对于死亡人数的异常波动,首先对死亡率进行一阶差分得到差分序列ΔD(x),然后用三次样条拟合差分序列,计算差分序列的趋势f(x)和标准偏差σ,若差分序列的某一点满足:|ΔD(x) -f(x)|>1.8σ,则认为该点为离群值,并将该点调整为:ΔD(x)′ =f(x)±1.8σ。详见Human Mortality Database,http://www.mortality.org/Public/Docs/MethodsProtocol.pdf.