7.6 Conclusion
8.1.2 Claim frequency
term chosen possibility
s df = 1
f leet df = 1
use df = 1
f uel df = 1
f uel·s df = 0
fc(cov) df = 2
gc(cov, s) df = 2
f1(ageph) df = 6
g1(ageph) df = 5
f2(agec) df = 10
g2(agec) df = 1
f3(hp) df = 7
g3(hp) df = 0
f4(bm) df = 14
g4(bm) df = 1
fspat(dist) df = 125 gspat(dist) df = 10
Table 8.4: Degrees of freedom chosen for the model of claim frequencies. For possible term types and possible degrees of freedom compare table 8.1.
2 8 10 9
12 11 13 15
16 17
1012 10 15
16
120120.4120.8AIC (in 1000)
0 1 2 3 4 5 6
iteration
Changes in AIC during the selection
Figure 8.7: Changes in AIC during the selection. The grey dots and numbers mark variables whose modelling is changed. The variables / terms belonging to the numbers are given in table 8.1.
and 80% confidence bands are shown in figure 8.8 and the average spatial effect together with 95% and 80% significance maps in figure 8.9. The sampling distributions of degrees of freedom obtained from bootstrapping can be found in figures 8.10 and 8.12. They can be used to perform a sensitivity analysis regarding the selected model.
Again, the selected model is similar to the model used by Denuit & Lang (2004). In
−.5−.250.25.5
20 40 60 80 100
policyholder’s age men women Effect: Policyholder’s age
−.5−.250.25.5
20 40 60 80
policyholder’s age Average effect: Policyholder’s age
−.5−.250.25.5
20 40 60 80
policyholder’s age Varying effect: Policyholder’s age
−.6−.30.3.6
0 4 8 12 16 20
car’s age men women Effect: Car’s age
−.6−.30.3.6
0 4 8 12 16 20
car’s age Average effect: Car’s age
−.6−.30.3.6
0 4 8 12 16 20
car’s age Varying effect: Car’s age
−1−.50.511.5
0 40 80 120 160 200 240
horsepower Average effect: Horsepower
−1−.50.511.5
0 40 80 120 160 200 240
horsepower Varying effect: Horsepower
−.6−.2.2.6
0 4 8 12 16 20
bonus−malus score men
women
Effect: Bonus−Malus Score
−.6−.2.2.6
0 4 8 12 16 20
bonus−malus score Average effect: Bonus−Malus Score
−.6−.2.2.6
0 4 8 12 16 20
bonus−malus score Varying effect: Bonus−Malus Score
Figure 8.8: Effects including confidence bands of the continuous covariates.
contrast to the model for claim sizes, there are more effects with an interaction regarding the gender of the policyholder. The policyholder’s age shows clearly different effects for men and women that were also discovered byDenuit & Lang (2004). Generally, young and
-0.4 0 0.6
average spatial effect significance map: average spatial effect
Figure 8.9: Average spatial effect and corresponding significance map. The significance map indicates significant positive (white or light grey) and significant negative regions (black or dark grey) at both 80% and 95% levels (white/black) or at 80% level (otherwise). The significance map for the varying spatial effect shows no variation and is therefore omitted.
old drivers produce more claims what is more clearly pronounced with men. Young and old women report less accidents than men of the same age whereas there is no difference between women and men for the age of 40 to 70. Note however, that both average and varying effect have broad confidence intervals for an age above 80 due to few observations in that range. The peak at an age of about 45 in the effects of both sexes could be caused by children driving their parent’s car. This peak is especially pronounced in the female effect what can be attributed to the fact that young car owners often ask their mother to purchase the policy (compare Denuit & Lang (2004)). The varying effect for the policyholder’s age is quite strong with the mode of the sampling distribution atdf = 3.
New cars produce more accidents than old cars. The effect reaches a local minimum at the age of three. This can be attributed to the Belgian characteristic that up to three year old cars don’t have to undergo the annual mechanical check–in. The male and female effects are nearly identical up to the age of three but differ afterwards: Women report less accidents than men. The number of accidents decreases for very old cars. Here, the varying effect is also identified as important with a mode at df = 1 corresponding to a linear varying effect.
The number of reported accidents increases with horsepower. Here, there is clearly no difference between the sexes. The effect of the bonus–malus score has also a positive trend but with differences between men and women: the effect is identical for values up to six, whereas for higher values women report less claims than men. The varying effect has its
0.05.1.15.2.25relative frequencies
75 80 85 90 95 100 105 110 115 120 125 130 Spatial effect: degrees of freedom
0.1.2.3.4.5.6relative frequencies
0 5 10 15 20
Spatial VC: degrees of freedom
0.2.4.6.81relative frequencies
cover cover*sex fleet fuel fuel*sex sex use
0 1 0 1 0 1 0 1 0 1 0 1 0 1
Fixed Effects: degrees of freedom
Figure 8.10: Sampling distributions of the different modelling alternatives obtained by boot-strap replications.
−.3−.2−.10.1.2.3
1 2 3
coverage
men women Effect: Coverage
Figure 8.11: Effect of coverage.
mode at df = 1 corresponding to a linear varying effect and is identified as important.
The average effect is very rough with a mode at df = 11 and a selected value of df = 14.
However, when estimating a model without the offset parameter risk, the selected value for the effect of bm isdf = 6 leading to a smooth, increasing function. (The modelling of all other terms is not influenced by removing the offset parameter.)
The spatial effect is also selected as varying over s, but the varying effect with a selected
value ofdf = 10 is only small. Moreover, the mode of the sampling distribution is atdf = 0 with a frequency of 60%. This indicates that the varying spatial effect is very uncertain and should rather be excluded from the model. The same is indicated by the significance maps (80% and 95%) that are zero everywhere (not shown). The average spatial effect shows that in urban areas more claims are reported and less claims in highly rural areas, especially the extreme south of Belgium. Hence, for claim frequencies the opposite effect can be observed compared to the claim size.
The effects of the categorical covariates are quite stable since the frequency distribution clearly support the selected alternatives. The only exception isusethat is selected with a frequency of only 60% indicating that the alternative of removing this variable from the model should be considered as well. The effect of coverage is here varying with s. As fc(cov) uses effect coding and gc(cov) dummy coding the marginal effects are obtained as
fc(f em)(cov) =
−γc1−γc2−γs , if cov = 1 γc1−γs , if cov = 2 γc2−γs , if cov = 3 fc(male)(cov) =
−γc1−γc2+γs , if cov = 1 γc1+γcs1+γs , if cov = 2 γc2+γcs2+γs , if cov = 3
For both sexes, the number of claims is largest for the simple alternativecov = 1. Women with comprehensive coverage (cov = 3) report more claims than withcov = 2 whereas the male effect shows no difference between these alternatives (compare figure8.11).
0.2.4.6.8relative frequencies
5 6 7 8 9 10 11 12 13
Effect − Policyholder’s age: degrees of freedom
0.1.2.3.4relative frequencies
1 2 3 4 5 6 7 8 9 10 11
VC − Policyholder’s age: degrees of freedom
0.1.2.3.4relative frequencies
7 8 9 10 11 12 13 14 15 16 17 18 Effect − Car’s age: degrees of freedom
0.2.4.6.8relative frequencies
0 1 2 3 4 5 6 7 8
VC − Car’s age: degrees of freedom
0.1.2.3.4relative frequencies
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Effect − Horsepower: degrees of freedom
0.2.4.6.81relative frequencies
0 1 2 3 4 5 6 7
VC − Horsepower: degrees of freedom
0.05.1.15.2.25relative frequencies
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Effect − Bonus−Malus Score: degrees of freedom
0.2.4.6.8relative frequencies
0 1 2 3 4 5 6
VC − Bonus−Malus Score: degrees of freedom
Figure 8.12: Sampling distributions of the different modelling alternatives obtained by boot-strap replications.