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1、准备金评估的准备金评估的GLM孟生旺中国人民大学统计学院1GLM模型的结构分布假设线性预测项连接函数2GLM模型的检验和比较关于解释变量：是否显著？方差分析，P值是否分段？是否线性？平滑函数是否存在交互效应？关于分布假设：残差分析：QQ图，蠕虫图。流量三角形的数据有限，分布假设的影响显著关于连接函数？模型比较：AIC=-2l +2*参数个数3准备金评估 的数据格式流量三角形数据框4事故年事故年进展年展年123456789101357,848766,940610,542482,940527,326574,398146,342139,950227,22967,9482352,118884,0219
2、33,8941,183,289445,745320,996527,804266,172425,0463290,5071,001,799926,2191,016,654750,816146,923495,992280,4054310,6081,108,250776,1891,562,400272,482352,053206,2865443,160693,190991,983769,488504,851470,6396396,132937,085847,498805,037705,9607440,832847,6311,131,3981,063,2698359,4801,061,6481,443,
3、3709376,686986,60810344,014增量赔款的流量三角形格式（增量赔款的流量三角形格式（Taylor，1983）5增量赔款的数据框格式增量赔款的数据框格式事故年事故年进展年进展年增量赔款增量赔款113578481276694013610542144829401552732616574398171463421813995019227229110679482135211822884021239338942411832892544574526320996275278042826617229425046312905076描述性统计分析78910第1个事故年的进展趋势比较特殊？11均值
4、和方差方差=方差/均值=195597Min.1stQu.MedianMean3rdQu.Max.67950352100527300624700905100156200012泊松分布拟合：图中泊松分布的概率被缩小为实际值的泊松分布拟合：图中泊松分布的概率被缩小为实际值的1%1314伽马分布拟合伽马分布拟合描述性分析的初步结论增量赔款右偏，方差远远大于均值，伽马拟合较好第1个事故年的进展模式与其它事故年不同15建模分布假设：泊松（链梯法）伽马解释变量：事故年、进展年、日历年形式：离散、连续连接函数：对数16泊松回归增量赔款 泊松分布log(增量赔款)=事故年+进展年1712345678910准备金
5、13578481,124,7881,735,3302,218,2702,745,5963,319,9943,466,3363,606,2863,833,5153,901,46323521181,236,1392,170,0333,353,3223,799,0674,120,0634,647,8674,914,0395,339,0855,433,71994,63432905071,292,3062,218,5253,235,1793,985,9954,132,9184,628,9104,909,3155,378,826469,51143106081,418,8582,195,0473,757,4
6、474,029,9294,381,9824,588,2685,297,906709,63854431601,136,3502,128,3332,897,8213,402,6723,873,3114,858,200984,88963961321,333,2172,180,7152,985,7523,691,7125,111,1711,419,45974408321,288,4632,419,8613,483,1305,660,7712,177,64183594801,421,1282,864,4986,784,7993,920,30193766861,363,2945,642,2664,278,
7、972103440144,969,8254,625,811进展因子3.49060651.7473331.4574131.1738521.1038241.0862691.0538741.0765551.01772473累积进展因子14.4465774.1387012.3685821.6251961.3844991.2542761.1546641.0956371.01772473准备金合计18,680,856链梯法（使用累积赔款数据）链梯法（使用累积赔款数据）18泊松回归的结果泊松回归的结果注意：泊松回归的所有变量高度显著！准备金估计值等价于链梯法，为18,680,85619问题？链梯法或泊松回归
8、适用于这组数据吗？需要进行模型的诊断和比较2021应用glm函数的残差分析泊松回归的随机分位残差：gamlss22泊松回归随机分位残差的QQ图：删除了无穷大的残差23结论泊松回归（链梯法）不适用于这组数据为什么所有变量是高度显著的？低估了标准误。改用伽马回归24伽马回归的结果25回归系数的比较泊松（泊松（链梯法）梯法）伽伽马(Intercept)12.50612.560factor(accyear)20.3310.317factor(accyear)30.3210.283factor(accyear)40.3060.165factor(accyear)50.2190.231factor(acc
9、year)60.2700.273factor(accyear)70.3720.352factor(accyear)80.5530.462factor(accyear)90.3690.307factor(accyear)100.2420.189factor(devyear)20.9130.909factor(devyear)30.9590.932factor(devyear)41.0260.998factor(devyear)50.4350.415factor(devyear)60.0800.111factor(devyear)7-0.006-0.054factor(devyear)8-0.39
10、4-0.450factor(devyear)90.009-0.059factor(devyear)10-1.380-1.43326标准误的比较（泊松回归系数为何高度显著？）27泊松回泊松回归归伽伽马马回回归归(Intercept)0.0007540.1568factor(accyear)20.00066940.1531factor(accyear)30.00068770.1601factor(accyear)40.00070080.1677factor(accyear)50.00073240.1768factor(accyear)60.00074450.1884factor(accyear)7
11、0.00076060.2041factor(accyear)80.00081330.2273factor(accyear)90.0010430.2673factor(accyear)100.0018640.3606factor(devyear)20.0006490.1531factor(devyear)30.00066520.1601factor(devyear)40.0006840.1677factor(devyear)50.00080190.1768factor(devyear)60.00093640.1884factor(devyear)70.0010390.2041factor(dev
12、year)80.0013530.2273factor(devyear)90.0013960.2673factor(devyear)100.003910.3606AIC的比较dfAIC泊松回泊松回归（链梯法）梯法）191903877伽伽马20150128伽马分布假设合适吗？残差分析29伽马回归的残差（glm）30伽马回归的残差（gamlss）31残差分析：伽马回归的蠕虫图32初步结论伽马回归优于泊松回归。伽马回归的分布假设通过检验。33伽马回归的偏残差34伽马回归能否进一步改进？使用平滑函数使用平滑函数：优点：可以提高预测结果的准确性缺点：增加解释困难3536完美拟合：折现完美拟合：折现完全平滑
13、：回归直线完全平滑：回归直线平滑函数的选择平滑函数的选择拟合效果好足够平滑惩罚样条惩罚样条：3738惩罚样条平滑惩罚样条平滑（0）log(增量赔款)=事故年+进展年（1）log(增量赔款)=f(事故年)+进展年（2）log(增量赔款)=事故年+f(进展年)（3）log(增量赔款)=f(事故年)+f(进展年)（4）log(增量赔款)=f(进展年)39伽马回归模型模型模型dfAIClog(增量增量赔款款)=f(事故年事故年)+进展年展年151492log(增量增量赔款款)=f(进展年展年)61496log(增量增量赔款款)=f(事故年事故年)+f(进展年展年)101496 log(增量增量赔款款)
14、=事故年事故年+进展年展年201501log(增量增量赔款款)=事故年事故年+f(进展年展年)151505伽马回归模型的比较伽马回归模型的比较4041如何比较如何比较AIC？与AIC的最小值之差模型的信息损失达到最小的相对概率10.60720.36830.22340.13550.08260.05070.03080.01890.011100.007平滑的伽马回归4243log(增量赔款)=f(事故年)+进展年44 Estimate Std.Error t value Pr(|t|)(Intercept)12.66056 0.12721 99.525|t|)(Intercept)13.48927
15、0.14075 95.836|t|)(Intercept)13.66911 0.08497 160.861|t|)(Intercept)12.795873 0.120328 106.341|t|)(Intercept)12.8163 0.0810 158.221|t|)(Intercept)13.54859 0.07867 172.226 2e-16*ps(devyear)-0.06337 0.01837 -3.449 0.00136*小结重视描述性分析用泊松回归检验链梯法的合理性如何改进链梯法的结果?伽马回归使用惩罚样条平滑函数对离散参数建模模型的评价和比较残差分析（分布假设）AIC（模型比较）53谢 谢！54