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

tel-00703797, version 2 - 7 Jun 2012
Chapitre 1. Sur la nécessité d’un modèle de marché
which emphasizes the exponential family characteristic. Let us recall the first two moments
are E(Yi) = πi and V ar(Yi) = πi(1 − πi) = V (πi). Assuming Yi is a Bernoulli distribution
B(πi) implies that πi is both the parameter and the mean value of Yi. So, the link function
for a Bernoulli model is expressed as follows
πi = g −1 (x T i β).
Let us note that if some individuals have identical covariates, then we can group the data
and consider Yi follows a binomial distribution B(ni, πi). However, this is only possible if all
covariates are categorical. As indicated in McCullagh and Nelder (1989), the link function and
the response variable can be reformulated in term of a latent variable approach πi = P (Yi =
1) = P (x T i β − ɛi > 0). If ɛi follows a normal distribution (resp. a logistic distribution), we
have πi = Φ(x T i β) (πi = Flogistic(x T i β)).
Now, the log-likelihood is derived as
ln(L(π1, . . . , πn, y1, . . . , yn)) =
n
[yi ln(πi) + (1 − yi) ln(1 − πi)] ,
plus an omitted term not involving πi. Further details can be found in Appendix 1.8.1.
Link functions
i=1
Generally, the following three functions are considered as link functions for the binary
variable
π
1. logit link: g(π) = ln 1−π with g−1 being the standard logistic distribution function,
2. probit link: g(π) = Φ −1 (π) with g −1 being the standard normal distribution function,
3. complementary log-log link: g(π) = ln(− ln(1−π)) with g −1 being the standard Gumbel
II distribution function ∗ .
On Figure 1.4 in Appendix 1.8.1, we plot these three link functions and their inverse functions.
All these three functions are the inverses of a distribution function, so other link functions can
be obtained using inverses of other distribution function. Let us note that the first two links
are symmetrical, while the last one is not.
In addition to being the canonical link function for which the fitting procedure is simplified,
cf. Appendix 1.8.1, the logit link is generally preferred because of its simple interpretation as
the logarithm of the odds ratio. Indeed, assume there is one explanatory variable X, the logit
link model is p/(1 − p) = e µ+αX . If ˆα = 2, increasing X by 1 will lead to increase the odds
by e 2 ≈ 7.389.
1.2.3 Variable selection and model adequacy
As fitting a GLM is quick in most standard software, then a relevant question is to check
for its validity on the dataset used.
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∗. A Gumbel of second kind is the distribution of −X when X follows a Gumbel distribution of first kind.