Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Efficiency and Bonus Hunger 235
Table 5.4 Results of the Inverse Gaussian regression on the claim costs recorded in Portfolio C.
Variable Level Coeff Std error Wald 95 % conf limit Chi-sq Pr>Chi-sq
Intercept 71169 00282 70616 71722 636343 < .0001
Ageph 18–24 02293 01104 00129 04457 431 0.0378
Ageph 25–30 01699 00697 00334 03065 595 0.0147
Ageph > 30 0 0 0 0 . .
Agev 0–2 01609 00756 00127 03091 453 0.0334
Agev > 2 0 0 0 0 . .
LogNormal Distribution
Before the generalized linear models gained popularity in the actuarial profession, claim sizes
were often analysed using a Normal linear regression model after having been transformed
to the log-scale. Although the results are usually quite similar for this method and Gamma
regression, the latter approach is easier to interpret since it does not require any logarithmic
transformation of the claim costs.
Assume that the moderate claim sizes for policyholder i are independent and LogNormally
distributed, with parameters 0 + ∑ p
j=1 jx ij and 2 . Specifically, the C ik s are independent,
and identically distributed for fixed i, with C ik ∼ Nor 0 + ∑ p
j=1 jx ij 2 . Let n i be the
number of claims reported by policyholder i, and let c i1 c i2 c ini be the corresponding
claim costs. The likelihood associated with the observations is
= ∏
n
∏ i
in i >0 k=1
⎛ (
1
√
2cik exp ⎝− 1
2 2
ln c ik − 0 −
p∑
j=1
j x ij
) 2
⎞
⎠
The maximum likelihood estimators are obtained with the help of Newton-Raphson
techniques. The average cost of a standard claim for policyholder i is then obtained from
the formula
(
)
p∑
EC ik x i = exp 0 + j x ij + 2
2
Note that, in contrast to the Gamma and Inverse Gaussian cases, we cannot easily deal with
the situation where only the total amount of moderate claims is available. This is due to the
fact that the LogNormal family of distributions is not closed under convolution. Therefore,
we fit the model to the observations made on policyholders having filed a single standard
claim.
Example 5.3 (LogNormal Regression for the Moderate Claim Costs in Portfolio C) The
LogNormal regression cannot be performed with the help of SAS R /STAT procedure
GENMOD (which does not support the LogNormal distribution). Often in practice, the
data are first transformed on the log-scale, and a standard linear model is then fitted to
the logarithms of the claim amounts. This ad-hoc procedure will be avoided here, and a
maximum likelihood estimation procedure is performed on the original claim costs.
j=1