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
η = g(π)
-5 0 5
Univariate smoothing
Link functions
logit
probit
compl. log-log
0.0 0.2 0.4 0.6 0.8 1.0
π
π = g(η) (−1)
0.0 0.2 0.4 0.6 0.8 1.0
Inverse link functions
inv. logit
inv. probit
inv. cloglog
-4 -2 0 2 4
Figure 1.4: Link functions for binary regression
In this paragraph, we present gradually some classic smoothing procedures, from the simplest
to more complex methods. Probably the simplest method to get a smooth function is to
regress a polynom on the whole data. Assuming observations are denoted by x1, . . . , xn and
y1, . . . , yn, a multiple regression model is appropriate with
Y = α0 + α1X + · · · + αpX p .
Using f(x) =
i αix i is clearly not flexible and a better tool has to be found. One way
to be more flexible in the smoothing is to subdivide the interval [min(x), max(x)] into K
segments. And then we can compute the average of the response variable Y on each segment
[ck, ck+1[. This is called the bin smoother in the literature. As shown on Hastie and Tibshirani
(1990) Figure 2.1, this smoother is rather unsmooth.
Another way to find a smooth value at x, we can use points about x, in a symmetric
neighborhood NS(x). Typically, we use the k nearest point at the left and k nearest at the
right of x to compute the average of yi’s. We have
s(y|x) =
1
CardNS(x)
i∈NS(x)
where the cardinal CardNS(x) does not necessarily equal to 2k + 1 if x is near the boundaries.
Again we do not show the result and refers the reader to Hastie and Tibshirani (1990) Figure
2.1. This method, called the running mean, takes better into account the variability of the
data. However we lose the smoothness of previous approches.
An extension of this approach is to fit the linear model y = µ + αx on the points (xi, yi)
in the neighborhood (for i ∈ NS(x)). That is to say we have a serie of intercepts µ and slopes
α for all observations. We called this method the running line, which generalizes the running
mean, where α is forced to 0.
76
yi,
η