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

tel-00703797, version 2 - 7 Jun 2012
1.2. GLMs, a brief introduction
2. a systematic component: the covariate vector Xi provides a linear predictor ∗ ηi = X T i β,
3. a link function: g : R ↦→ S which is monotone, differentiable and invertible, such that
E(Yi) = g −1 (ηi),
for all individuals i ∈ {1, . . . , n}, where θi is the shape parameter, φi the dispersion parameter,
a, b, c three functions and S a set of possible values of the expectation E(Yi). Let us note
that we get back to linear models with a Gaussian distribution and an identity link function
(g(x) = x). However, there are many other distributions and link functions. We say a link
function to be canonical if θi = ηi.
Distribution Canonical link Mean Purpose
Normal N (µ, σ 2 ) identity: ηi = µi µ = Xβ standard linear regression
Bernoulli B(µ) logit: ηi = log( µ
1−µ ) µ = 1
1+e −Xβ
rate modelling
Poisson P(µ) log: ηi = log(µi) µ = e Xβ claim frequency
Gamma G(α, β) inverse: ηi = 1
µi
Inverse Normal I(µ, λ) squared inverse: ηi = − 1
µ 2 i
Table 1.1: Family and link functions
µ = (Xβ) −1 claim severity
µ = (Xβ) −2 claim severity
There are many applications of GLM in actuarial science. Table 1.1 below lists the most
common distributions with their canonical link and standard applications. Apart from the
identity link function, the log link function is the most commonly used link function in actuarial
applications. In fact, with this link function, the explanatory variables have multiplicative
effects on the observed variable and the observed variable stays positive, since E(Y ) =
i eβixi .
For example, the effect of being a young driver and owning an expensive car on average loss
could be the product of the two separate effects: the effect of being a young driver and the
effect of owning an expensive car. The logarithm link function is a key element in most
actuarial pricing models and is used for modelling the frequency and the severity of claims.
It makes possible to have a standard premium and multiplicative individual factors to adjust
the premium.
1.2.2 Binary regression
Since the insurer choice is a Bernoulli variable, we give further details on binary regression
in this subsection.
Base model assumption
In binary regression, the response variable is either 1 or 0 for success and failure, respectively.
We cannot parametrize two outcomes with more than one parameter. So, a Bernoulli
distribution B(πi) is assumed, i.e. P (Yi = 1) = πi = 1 − P (Yi = 0), with πi the parameter.
The mass probability function can be expressed as
fYi (y) = πy
i (1 − πi) 1−y ,
∗. For GLMs, the name ‘linear predictor’ is kept, despite ηi is not a linear predictor of Yi.
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