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
where Θ denotes the (unknown) parameter vector and E the (random) noise vector. The
linear model assumptions are: (i) white noise: E(Ei) = 0, (ii) homoskedasticity: V ar(Ei) =
σ 2 , (iii) normality: Ei ∼ N (0, σ 2 ), (iv) independence: Ei is independent of Ej for i = j,
(v) parameter identification: rank(X) = p < n. Then, the Gauss-Markov theorem gives
us the following results: (i) the least square estimator ˆ Θ of Θ is ˆ Θ = (X T X) −1 X T Y and
ˆσ 2 = ||Y − XΘ|| 2 /(n − p) for σ 2 , (ii) ˆ Θ is a Gaussian vector independent of the random
variable ˆσ 2 ∼ χ 2 n−p, (iii) ˆ Θ is an unbiased estimator with minimum variance of Θ, such that
V ar( ˆ Θ) = σ 2 (X T X) −1 and ˆσ 2 is an unbiased estimator of σ 2 .
Let us note that first four assumptions can be expressed into one single assumption
E ∼ N (0, σ 2 In). But splitting the normality assumption will help us to identify the strong
differences between linear models and GLMs. The term XΘ is generally referred to the linear
predictor of Y .
Linear models include a wide range of statistical models, e.g. the simple linear regression
yi = a + bxi + ɛi is obtained with a 2-column matrix X having 1 in first column and (xi)i
in second column. Many properties can be derived for linear models, notably hypothesis
tests, confidence intervals for parameter estimates as well as estimator convergence, see, e.g.,
Chapter 6 of Venables and Ripley (2002).
We now focus on the limitations of linear models resulting from the above assumptions.
The following problems have been identified. When X contains near-colinear variables, the
computation of the estimator Θ will be numerically unstable. This would lead to an increase
in the variance estimator ∗ . Working with a constrained linear model is not an appropriate
answer. In pratice, a solution is to test models with omitting one explanatory variable after
another to check for near colinearity. Another stronger limitation lies in the fact that the
response variance is assumed to be the same (σ 2 ) for all individuals. One way to deal with this
issue is to transform the response variable by the nonlinear Box-Cox transformation. However,
this response transformation can still be unsatifactory in certain cases. Finally, the strongest
limitation is the assumed support of the response variable. By the normal assumption, Y
must lies in the whole set R, which excludes count variable (e.g. Poisson distribution) or
positive variable (e.g. exponential distribution). To address this problem, we have to use a
more general model than linear models.
In this paper, Y represents the lapse indicator of customers, i.e. Y follows a Bernoulli
variable with 1 indicating a lapse. For Bernoulli variables, there are two main pitfalls. Since
the value of E(Y ) is contained within the interval [0, 1], it seems natural the expected values ˆ Y
should also lie in [0, 1]. However, predicted values θX may fall out of this range for sufficiently
large or small values of X. Furthermore, the normality hypothesis of the residuals is clearly
not met: Y − E(Y ) will only take two different values, −E(Y ) and 1 − E(Y ). Therefore, the
modelling of E(Y ) as a function of X needs to be changed as well as the error distribution.
This motivates to use an extended model that can deal with discrete-valued variables.
Toward generalized linear models
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A Generalized Linear Model is characterized by three components:
1. a random component: Yi follows a specific distribution of the exponential family Fexp(θi, φi, a, b, c) † ,
∗. This would be one way to detect such isssue.
†. See Appendix 1.8.1.