Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
function is given by
P (Y = k) =
π if k = 0,
1.5. Testing asymmetry of information
(1 − π) λk
k! e−λ otherwise.
Note that Y is a mixture of a Bernoulli distribution B(π) with a Poisson distribution P(λ).
The mean of the zero-inflated Poisson distribution is (1 − π)λ. Using the GLM framework
and the canonical link functions, a zero-inflated GLM Poisson model is defined as
E(Y/ xi) =
1
1 + e x1 T e
i γ x2 i
where the covariate vectors x 1 i , x2 i are parts of the vector xi. Now there are two (vector)
coefficients to estimate β and γ. The GLM is implemented in R base by the glm function.
For the zero-inflated model, we need to use the pscl package, cf. Jackman (2011).
Still studying the FC agent dataset, we fit three distributions on the claim number: Poisson,
zero-inflated Poisson and Negative binomial distributions. As shown in Table 1.19 in
Appendix 1.8.2, the three models are similar in terms of log-likelihood or AIC. But, differences
appear at the predictions.
Despite being equivalent for first probabilities P (X = 0, 1, 2), classic and zero-inflated
Poisson distributions decrease too sharply compared to the observed number of claims. The
negative Binomial distribution (fourth line) is far better. In Appendix 1.8.2, we give the
regression summary for zero-inflated negative binomial distribution on the FC agent subset.
We obtain the same conclusion for other FC subsets.
Claim number 0 1 2 3 4 5 6
Observed 43687 5308 667 94 17 2 2
Poisson 43337.9 5896.0 500.9 39.8 3.7 0.417 0.054
zeroinfl. Poisson 43677.6 5267.7 745.0 80.2 7.5 0.665 0.058
zeroinfl. NB 43704.6 5252.6 704.7 98.8 14.9 2.457 0.442
Table 1.15: Claim number prediction for Full Comp. agent subset
Testing adverse selection
Now that we have modelled the claim frequency, we turn to the modelling of the deductible
choice as described in the previous section: an ordered logistic model. We test for evidence of
adverse selection on three datasets: agent, broker and direct with Full. Comp. products. Let
us note that we cannot test adverse selection on TPL covers, since there is no deductible for this
cover. As reported in Subsection 1.5.1, adverse selection testing is done by a fit of a GLM to
explain the deductible choice Zi. In addition to the exogeneous variables Xi for ith individual,
the regression will use the observed claim number Yi (endogeneous) and its expected value
coming from the zero-inflated negative binomial regression E(Y |Xi) (exogeneous).
The numerical illustrations reveal that it is more relevant to cluster some deductible values
which are too few in the dataset. Actually, the deductible is valued in {0, 150, 300, 500, 600,
1000, 2000, 2500}. As 300 euros is the standard deductible, very high deductibles are rarely
chosen. So, we choose to regroup deductible values greater than 500 together. In Table 1.16,
T β,
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