索赔次数的开放式混合泊松分布研究
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摘要
本文建立了索赔次数的多风险类别混合泊松分布。首先,考虑索赔次数的零膨胀、厚尾性和异质性等特征,建立风险类别待定的开放式混合泊松分布(OMP分布),开放式结构使该分布对实际数据的多样特征和风险类别具有良好的自适应性;其次,定义混合权重参数的iSCAD惩罚函数,实现对权重参数的筛选;最后,借助EM算法求得分布参数,实现对各风险类别下索赔次数的估计。借助iSCAD惩罚函数,本文给出最优混合数,避免传统混合分布中主观选择的弊端,克服传统混合分布中结构复杂、参数估计没有显式表达式、估计结果不便于解释等问题。基于三组风险特征多样数据的实证分析,本文发现OMP分布可以显著改进现有模型的拟合效果。
This paper builds up an open mixed Poisson model with multiple risk categories of claim frequency. Firstly, an open mixed Poisson model is set up with risk categories to be defined, taking into account the characteristics, such as zero-inflated, heavy tailing and heterogeneity of the claim frequency. The open structure makes the model self-adaptable to various characteristics and risk categories in the actual data. Secondly, iSCAD penalty function defined with mixed weight parameters is applied to choose the suitable parameters. And finally, EM algorithm is used to acquire all the estimates of the claim frequency in different risk categories. By means of the iSCAD penalty function, the optimal mixed number is derived, keeping away from the subjective selection in the traditional mixed model, and solving the issues such as complicated structures, no explicit expression of parameter estimates and difficulty in explaining the estimates in the traditional mixed models. Based on the real data in three risk categories, the empirical study finds that the new model can significantly improve the fitting effects of the traditional ones.
This paper builds up an open mixed Poisson model with multiple risk categories of claim frequency. Firstly, an open mixed Poisson model is set up with risk categories to be defined, taking into account the characteristics, such as zero-inflated, heavy tailing and heterogeneity of the claim frequency. The open structure makes the model self-adaptable to various characteristics and risk categories in the actual data. Secondly, iSCAD penalty function defined with mixed weight parameters is applied to choose the suitable parameters. And finally, EM algorithm is used to acquire all the estimates of the claim frequency in different risk categories. By means of the iSCAD penalty function, the optimal mixed number is derived, keeping away from the subjective selection in the traditional mixed model, and solving the issues such as complicated structures, no explicit expression of parameter estimates and difficulty in explaining the estimates in the traditional mixed models. Based on the real data in three risk categories, the empirical study finds that the new model can significantly improve the fitting effects of the traditional ones.
引文
[1]Aryuyuen S, Bodhisuwan W. The Negative Binomial-generalized Exponential (Nb-Ge) Distribution[J]. Applied Mathematical Sciences, 2013(7):1093-1105.
[2]毛泽春, 刘锦萼. 指数类混合型索赔次数的分布及其应用[J].应用概率统计,2008(24):1-11.
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[6]杨亮, 孟生旺. 零膨胀损失次数的贝叶斯分位回归模型[J]. 数量经济技术经济研究, 2017(34):149-160.
[7]Famoye F, Singh K P. Zero-inflated Generalized Poisson Regression Model with an Application to Domestic Violence Data[J]. Journal of Data Science, 2006(4):117-130.
[8]Shoukri M, et al. The Poisson Inverse Gaussian Regression Model in the Analysis of Clustered Counts Data[J]. Journal of Data Science, 2004(2):17-32.
[9]Chen J, Khalili A. Order Selection in Finite Mixture Models with a Nonsmooth Penalty[J]. Journal of the American Statistical Association, 2008(103):1674-1683.
[10]Lee S C, Lin X S. Modeling and Evaluating Insurance Losses Via Mixtures of Erlang Distributions[J]. North American Actuarial Journal, 2010(14):107-130.
[11]Verbelen R, et al. Fitting Mixtures of Erlangs to Censored and Truncated Data Using the Em Algorithm[J]. ASTIN Bulletin: The Journal of the IAA, 2015(45):729-758.
[12]Yin C, Lin X S. Efficient Estimation of Erlang Mixtures Using Iscad Penalty with Insurance Application[J]. ASTIN Bulletin: The Journal of the IAA, 2016(46):779-799.
[13]Mclachlan G J, Krishnan T. The EM Algorithm and Extensions[M].Second Edition.New York:John Wiley&Sons Inc,2008, 61-64.
[14]Klugman S A, Panjer H H, Willmot G E. Loss Models: From Data to Decisions[M]. Second Edition.New York:John Wiley&Sons Inc, 2012, 349-349.
[15]Zeileis A, Kleiber C, Jackman S. Regression Models for Count Data in R[J]. Journal of Statistical Software, 2008(27):1-25.
[16]Simon L R J. Fitting Negative Binomial Distributions by the Method of Maximum Likelihood[C].Proceedings of the Casualty Actuarial Society, 1961(48): 45-53.
[2]毛泽春, 刘锦萼. 指数类混合型索赔次数的分布及其应用[J].应用概率统计,2008(24):1-11.
[3]Joe H, Zhu R. Generalized Poisson Distribution: the Property of Mixture of Poisson and Comparison with Negative Binomial Distribution[J]. Biometrical Journal, 2005(47):219-229.
[4]毛泽春, 刘锦萼. 免赔额和NCD赔付条件下保险索赔次数的分布[J].中国管理科学,2005(5):1-5.
[5]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(11):7-102.
[6]杨亮, 孟生旺. 零膨胀损失次数的贝叶斯分位回归模型[J]. 数量经济技术经济研究, 2017(34):149-160.
[7]Famoye F, Singh K P. Zero-inflated Generalized Poisson Regression Model with an Application to Domestic Violence Data[J]. Journal of Data Science, 2006(4):117-130.
[8]Shoukri M, et al. The Poisson Inverse Gaussian Regression Model in the Analysis of Clustered Counts Data[J]. Journal of Data Science, 2004(2):17-32.
[9]Chen J, Khalili A. Order Selection in Finite Mixture Models with a Nonsmooth Penalty[J]. Journal of the American Statistical Association, 2008(103):1674-1683.
[10]Lee S C, Lin X S. Modeling and Evaluating Insurance Losses Via Mixtures of Erlang Distributions[J]. North American Actuarial Journal, 2010(14):107-130.
[11]Verbelen R, et al. Fitting Mixtures of Erlangs to Censored and Truncated Data Using the Em Algorithm[J]. ASTIN Bulletin: The Journal of the IAA, 2015(45):729-758.
[12]Yin C, Lin X S. Efficient Estimation of Erlang Mixtures Using Iscad Penalty with Insurance Application[J]. ASTIN Bulletin: The Journal of the IAA, 2016(46):779-799.
[13]Mclachlan G J, Krishnan T. The EM Algorithm and Extensions[M].Second Edition.New York:John Wiley&Sons Inc,2008, 61-64.
[14]Klugman S A, Panjer H H, Willmot G E. Loss Models: From Data to Decisions[M]. Second Edition.New York:John Wiley&Sons Inc, 2012, 349-349.
[15]Zeileis A, Kleiber C, Jackman S. Regression Models for Count Data in R[J]. Journal of Statistical Software, 2008(27):1-25.
[16]Simon L R J. Fitting Negative Binomial Distributions by the Method of Maximum Likelihood[C].Proceedings of the Casualty Actuarial Society, 1961(48): 45-53.