相依风险的团体健康保险损失预测
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摘要
在团体健康保险中,同一份保单通常包含若干被保险人,被保险人之间相依的风险特征使得保单的赔付数据呈现出分层结构的特点。同时,保单的索赔次数和索赔强度通常存在一定的相依关系,这种相依关系对保险公司纯保费的厘定结果具有重要的影响。为了准确预测团体健康保险的纯保费,本文建立了相依风险的贝叶斯分层模型,该模型用伽马分布来描述索赔强度数据,用零截断泊松分布来描述索赔次数数据,分别在模型均值中引入共同的随机效应来描述赔付数据的分层特征和索赔次数与索赔强度之间的相依关系;最后借助贝叶斯HMC算法进行参数估计,并给出了团体保单的损失预测分布。本文将该方法运用到我国一组团体健康保险的损失数据并对保单累积损失进行预测。结果表明,相依风险的贝叶斯分层模型具有良好的应用价值。
In the group health insurance, the policy usually contains several insureds. As the group insureds share similar risk characteristics, the claim data shows the hierarchical feature. Considering the dependence between numbers of claim and claim severity, this paper presents a Bayesian hierarchical model to predict the total loss. This model first assumes the numbers of claim follow the zero-truncated Poisson distribution and claim severity follows the gamma distribution, and then introduces the common random effect into the model to taking into account both the dependence between claim numbers and claim severity and the hierarchical structure of claim data. Finally, Bayesian approach is used to estimate the parameters and derive the predictive distribution of total loss. Numerical analysis is performed with claim data of the group health insurance from an insurance company, and the result shows that the model is appropriate to predict the total loss.
In the group health insurance, the policy usually contains several insureds. As the group insureds share similar risk characteristics, the claim data shows the hierarchical feature. Considering the dependence between numbers of claim and claim severity, this paper presents a Bayesian hierarchical model to predict the total loss. This model first assumes the numbers of claim follow the zero-truncated Poisson distribution and claim severity follows the gamma distribution, and then introduces the common random effect into the model to taking into account both the dependence between claim numbers and claim severity and the hierarchical structure of claim data. Finally, Bayesian approach is used to estimate the parameters and derive the predictive distribution of total loss. Numerical analysis is performed with claim data of the group health insurance from an insurance company, and the result shows that the model is appropriate to predict the total loss.
引文
[1]孟生旺,邱子真.混合效应模型及其在非寿险费率厘定中的应用[J].数理统计与管理,2016,35(1):154-161.
[2]Stasinopoulos D M,Rigby R A.Generalized additive models for location scale and shape(GAMLSS)in R[J].Journal of Statistical Software,2007,23(7):1-46.
[3]Fellingham G W,Kottas A,Hartman B M.Bayesian nonparametric predictive modeling of group health claims[J].Insurance:Mathematics and Economics,2015,60:1-10.
[4]Frees E W,Valdez E A.Hierarchical insurance claims modeling[J].Journal of the American Statistical Association,2008,103(484):1457-1469.
[5]Klein N,Kneib T,Lang S.Bayesian generalized additive models for location,scale,and shape for zero-inflated and overdispersed count data[J].Journal of the American Statistical Association,2015,110(509):405-419.
[6]孟生旺,杨亮.随机效应零膨胀索赔次数回归模型[J].统计研究,2015,32(11):97-103.
[7]孟生旺,李政宵.偏t分布假设下的空间效应模型及其应用[J].数理统计与管理,2016,35(6):1028-1037.
[8]Shi P,Feng X,Ivantsova A.Dependent frequency-severity modeling of insurance claims[J].Insurance:Mathematics and Economics,2015,64:417-428.
[9]孟生旺,李政宵.索赔频率与索赔强度的相依性模型[J].统计研究,2017,34(1):55-66.
[10]Gschl(o|¨)β1 S,Czado C.Spatial modelling of claim frequency and claim size in non-life insurance[J].Scandinavian Actuarial Journal,2007,(3):202-225.
[11]Kramer N,Brechmann E C,Silvestrini D,Czado C.Total loss estimation using copula-based regression models[J].Insurance:Mathematics and Economics,2013,53(3):829-839.
[12]Baumgartner C,Gruber L F,Czado C.Bayesian total loss estimation using shared random effects[J].Insurance:Mathematics and Economics,2015,62:194-201.
[13]Vehtari A,Gelman A,Gabry J.Efficient implementation of leave-one-out cross-validation and WAIC for evaluating fitted Bayesian models[R].arXiv preprint:arXiv:150704544,2015.
[14]Shi P,Basu S,Meyers G G.A Bayesian log-normal model for multivariate loss reserving[J].North American Actuarial Journal,2012,16(1):29-51.
[15]Gelman A.Prior distributions for variance parameters in hierarchical models(comment on article by Browne and Draper)[J].Bayesian analysis,2006,1(3):515-534.
[16]Gelman A,Jakulin A,Pittau M G,Su Y S.A weakly informative default prior distribution for logistic and other regression models[J].The Annals of Applied Statistics,2008:1360-1383.
[17]Girolami M,Calderhead B.Riemann manifold langevin and hamiltonian monte carlo methods[J].Journal of the Royal Statistical Society:Statistical Methodology(Series B),2011,73(2):123-214.
[18]Gelman A,Rubin D B.Inference from iterative simulation using multiple sequences[J].Statistical science,1992:457-72.
[19]Robert C P,Casella G.Monte Carlo Statistical Methods[M].New York:Springer,2005:499-500.
[20]Gelman A,Carlin J B,Stern H S,Rubin D B.Bayesian Data Analysis[M].Boca Raton,FL:Chapman&Hall/CRC,2014.
[2]Stasinopoulos D M,Rigby R A.Generalized additive models for location scale and shape(GAMLSS)in R[J].Journal of Statistical Software,2007,23(7):1-46.
[3]Fellingham G W,Kottas A,Hartman B M.Bayesian nonparametric predictive modeling of group health claims[J].Insurance:Mathematics and Economics,2015,60:1-10.
[4]Frees E W,Valdez E A.Hierarchical insurance claims modeling[J].Journal of the American Statistical Association,2008,103(484):1457-1469.
[5]Klein N,Kneib T,Lang S.Bayesian generalized additive models for location,scale,and shape for zero-inflated and overdispersed count data[J].Journal of the American Statistical Association,2015,110(509):405-419.
[6]孟生旺,杨亮.随机效应零膨胀索赔次数回归模型[J].统计研究,2015,32(11):97-103.
[7]孟生旺,李政宵.偏t分布假设下的空间效应模型及其应用[J].数理统计与管理,2016,35(6):1028-1037.
[8]Shi P,Feng X,Ivantsova A.Dependent frequency-severity modeling of insurance claims[J].Insurance:Mathematics and Economics,2015,64:417-428.
[9]孟生旺,李政宵.索赔频率与索赔强度的相依性模型[J].统计研究,2017,34(1):55-66.
[10]Gschl(o|¨)β1 S,Czado C.Spatial modelling of claim frequency and claim size in non-life insurance[J].Scandinavian Actuarial Journal,2007,(3):202-225.
[11]Kramer N,Brechmann E C,Silvestrini D,Czado C.Total loss estimation using copula-based regression models[J].Insurance:Mathematics and Economics,2013,53(3):829-839.
[12]Baumgartner C,Gruber L F,Czado C.Bayesian total loss estimation using shared random effects[J].Insurance:Mathematics and Economics,2015,62:194-201.
[13]Vehtari A,Gelman A,Gabry J.Efficient implementation of leave-one-out cross-validation and WAIC for evaluating fitted Bayesian models[R].arXiv preprint:arXiv:150704544,2015.
[14]Shi P,Basu S,Meyers G G.A Bayesian log-normal model for multivariate loss reserving[J].North American Actuarial Journal,2012,16(1):29-51.
[15]Gelman A.Prior distributions for variance parameters in hierarchical models(comment on article by Browne and Draper)[J].Bayesian analysis,2006,1(3):515-534.
[16]Gelman A,Jakulin A,Pittau M G,Su Y S.A weakly informative default prior distribution for logistic and other regression models[J].The Annals of Applied Statistics,2008:1360-1383.
[17]Girolami M,Calderhead B.Riemann manifold langevin and hamiltonian monte carlo methods[J].Journal of the Royal Statistical Society:Statistical Methodology(Series B),2011,73(2):123-214.
[18]Gelman A,Rubin D B.Inference from iterative simulation using multiple sequences[J].Statistical science,1992:457-72.
[19]Robert C P,Casella G.Monte Carlo Statistical Methods[M].New York:Springer,2005:499-500.
[20]Gelman A,Carlin J B,Stern H S,Rubin D B.Bayesian Data Analysis[M].Boca Raton,FL:Chapman&Hall/CRC,2014.
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