This week, we started to discuss overdispersion when modeling claims frequency. In my previous post, I discussed computations of empirical variances with different exposure. But I did use only one factor to compute classes. Of course, it is possible to use much more factors. For instance, using cartesian products of factors,
> X=as.factor(paste(sinistres$carburant,sinistres$zone, + cut(sinistres$ageconducteur,breaks=c(17,24,40,65,101)))) > E=sinistres$exposition > Y=sinistres$nbre > vm=vv=ve=rep(NA,length(levels(X))) > for(i in 1:length(levels(X))){ + ve[i]=Ei=E[X==levels(X)[i]] + Yi=Y[X==levels(X)[i]] + vm[i]=meani=weighted.mean(Yi/Ei,Ei) # moyenne + vv[i]=variancei=sum((Yi-meani*Ei)^2)/sum(Ei) # variance + cat("Class ",levels(X)[i],"average =",meani," variance =",variancei,"\n") + } Class D A (17,24] average = 0.06274415 variance = 0.06174966 Class D A (24,40] average = 0.07271905 variance = 0.07675049 Class D A (40,65] average = 0.05432262 variance = 0.06556844 Class D A (65,101] average = 0.03026999 variance = 0.02960885 Class D B (17,24] average = 0.2383109 variance = 0.2442396 Class D B (24,40] average = 0.06662015 variance = 0.07121064 Class D B (40,65] average = 0.05551854 variance = 0.05543831 Class D B (65,101] average = 0.0556386 variance = 0.0540786 Class D C (17,24] average = 0.1524552 variance = 0.1592623 Class D C (24,40] average = 0.0795852 variance = 0.09091435 Class D C (40,65] average = 0.07554481 variance = 0.08263404 Class D C (65,101] average = 0.06936605 variance = 0.06684982 Class D D (17,24] average = 0.1584052 variance = 0.1552583 Class D D (24,40] average = 0.1079038 variance = 0.121747 Class D D (40,65] average = 0.06989518 variance = 0.07780811 Class D D (65,101] average = 0.0470501 variance = 0.04575461 Class D E (17,24] average = 0.2007164 variance = 0.2647663 Class D E (24,40] average = 0.1121569 variance = 0.1172205 Class D E (40,65] average = 0.106563 variance = 0.1068348 Class D E (65,101] average = 0.1572701 variance = 0.2126338 Class D F (17,24] average = 0.2314815 variance = 0.1616788 Class D F (24,40] average = 0.1690485 variance = 0.1443094 Class D F (40,65] average = 0.08496827 variance = 0.07914423 Class D F (65,101] average = 0.1547769 variance = 0.1442915 Class E A (17,24] average = 0.1275345 variance = 0.1171678 Class E A (24,40] average = 0.04523504 variance = 0.04741449 Class E A (40,65] average = 0.05402834 variance = 0.05427582 Class E A (65,101] average = 0.04176129 variance = 0.04539265 Class E B (17,24] average = 0.1114712 variance = 0.1059153 Class E B (24,40] average = 0.04211314 variance = 0.04068724 Class E B (40,65] average = 0.04987117 variance = 0.05096601 Class E B (65,101] average = 0.03123003 variance = 0.03041192 Class E C (17,24] average = 0.1256302 variance = 0.1310862 Class E C (24,40] average = 0.05118006 variance = 0.05122782 Class E C (40,65] average = 0.05394576 variance = 0.05594004 Class E C (65,101] average = 0.04570239 variance = 0.04422991 Class E D (17,24] average = 0.1777142 variance = 0.1917696 Class E D (24,40] average = 0.06293331 variance = 0.06738658 Class E D (40,65] average = 0.08532688 variance = 0.2378571 Class E D (65,101] average = 0.05442916 variance = 0.05724951 Class E E (17,24] average = 0.1826558 variance = 0.2085505 Class E E (24,40] average = 0.07804062 variance = 0.09637156 Class E E (40,65] average = 0.08191469 variance = 0.08791804 Class E E (65,101] average = 0.1017367 variance = 0.1141004 Class E F (17,24] average = 0 variance = 0 Class E F (24,40] average = 0.07731177 variance = 0.07415932 Class E F (40,65] average = 0.1081142 variance = 0.1074324 Class E F (65,101] average = 0.09071118 variance = 0.1170159
Again, one can plot the variance against the average,
> plot(vm,vv,cex=sqrt(ve),col="grey",pch=19, + xlab="Empirical average",ylab="Empirical variance") > points(vm,vv,cex=sqrt(ve)) > abline(a=0,b=1,lty=2)
An alternative is to use a tree. The tree can be obtained from another variable (the insured had, or had not, a claim, during the period considered) but it should be rather close to the one we would like to model (the number of claims over the period considered). Here, I did use the whole database (with more that 600,000 lines)
> library(tree) > T=tree((nombre>0)~as.factor(zone)+as.factor(puissance)+ + as.factor(marque)+as.factor(carburant)+as.factor(region)+ + agevehicule+ageconducteur,data=baseFREQ, + split = "gini",minsize =25000)
The tree is the following
> plot(T) > text(T)
Now, each knot defines a class, and it is possible to use it to define a class. Which is supposed to be homogeneous.
> X=as.factor(T$where) > E=sinistres$exposition > Y=sinistres$nbre > vm=vv=ve=rep(NA,length(levels(X))) > for(i in 1:length(levels(X))){ + ve[i]=Ei=E[X==levels(X)[i]] + Yi=Y[X==levels(X)[i]] + vm[i]=meani=weighted.mean(Yi/Ei,Ei) # moyenne + vv[i]=variancei=sum((Yi-meani*Ei)^2)/sum(Ei) # variance + cat("Class ",levels(X)[i],"average =",meani," variance =",variancei,"\n") + } Class 6 average = 0.04010406 variance = 0.04424163 Class 8 average = 0.05191127 variance = 0.05948133 Class 9 average = 0.07442635 variance = 0.08694552 Class 10 average = 0.4143646 variance = 0.4494002 Class 11 average = 0.1917445 variance = 0.1744355 Class 15 average = 0.04754595 variance = 0.05389675 Class 20 average = 0.08129577 variance = 0.0906322 Class 22 average = 0.05813419 variance = 0.07089811 Class 23 average = 0.06123807 variance = 0.07010473 Class 24 average = 0.06707301 variance = 0.07270995 Class 25 average = 0.3164557 variance = 0.2026906 Class 26 average = 0.08705041 variance = 0.108456 Class 27 average = 0.06705214 variance = 0.07174673 Class 30 average = 0.05292652 variance = 0.06127301 Class 31 average = 0.07195285 variance = 0.08620593 Class 32 average = 0.08133722 variance = 0.08960552 Class 34 average = 0.1831559 variance = 0.2010849 Class 39 average = 0.06173885 variance = 0.06573939 Class 41 average = 0.07089419 variance = 0.07102932 Class 44 average = 0.09426152 variance = 0.1032255 Class 47 average = 0.03641669 variance = 0.03869702 Class 49 average = 0.0506601 variance = 0.05089276 Class 50 average = 0.06373107 variance = 0.06536792 Class 51 average = 0.06762947 variance = 0.06926191 Class 56 average = 0.06771764 variance = 0.07122379 Class 57 average = 0.04949142 variance = 0.05086885 Class 58 average = 0.2459016 variance = 0.2451116 Class 59 average = 0.05996851 variance = 0.0615773 Class 61 average = 0.07458053 variance = 0.0818608 Class 63 average = 0.06203737 variance = 0.06249892 Class 64 average = 0.07321618 variance = 0.07603106 Class 66 average = 0.07332127 variance = 0.07262425 Class 68 average = 0.07478147 variance = 0.07884597 Class 70 average = 0.06566728 variance = 0.06749411 Class 71 average = 0.09159605 variance = 0.09434413 Class 75 average = 0.03228927 variance = 0.03403198 Class 76 average = 0.04630848 variance = 0.04861813 Class 78 average = 0.05342351 variance = 0.05626653 Class 79 average = 0.05778622 variance = 0.05987139 Class 80 average = 0.0374993 variance = 0.0385351 Class 83 average = 0.06721729 variance = 0.07295168 Class 86 average = 0.09888492 variance = 0.1131409 Class 87 average = 0.1019186 variance = 0.2051122 Class 88 average = 0.05281703 variance = 0.0635244 Class 91 average = 0.08332136 variance = 0.09067632 Class 96 average = 0.07682093 variance = 0.08144446 Class 97 average = 0.0792268 variance = 0.08092019 Class 99 average = 0.1019089 variance = 0.1072126 Class 100 average = 0.1018262 variance = 0.1081117 Class 101 average = 0.1106647 variance = 0.1151819 Class 103 average = 0.08147644 variance = 0.08411685 Class 104 average = 0.06456508 variance = 0.06801061 Class 107 average = 0.1197225 variance = 0.1250056 Class 108 average = 0.0924619 variance = 0.09845582 Class 109 average = 0.1198932 variance = 0.1209162
Here, when ploting the empirical variance (per knot) against the empirial average of claims, we get
Here, we can identify classes where remaining heterogeneity.
Arthur Charpentier
Arthur Charpentier, professor at UQaM in Actuarial Science. Former professor-assistant at ENSAE Paristech, associate professor at Ecole Polytechnique and assistant professor in Economics at Université de Rennes 1. Graduated from ENSAE, Master in Mathematical Economics (Paris Dauphine), PhD in Mathematics (KU Leuven), and Fellow of the French Institute of Actuaries.
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