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

tel-00703797, version 2 - 7 Jun 2012
s(speed,1.49)
s(speed,1.53)
-40 -20 0 20 40
-40 -20 0 20 40
Thin plate regression splines (GCV)
5 10 15 20 25
speed
Thin plate regression splines (REML)
5 10 15 20 25
speed
(default method)
s(speed,1.49)
s(speed,1.53)
-40 -20 0 20 40
-40 -20 0 20 40
Cubic regression splines (GCV)
5 10 15 20 25
speed
Cubic regression splines (REML)
5 10 15 20 25
speed
Figure 1.7: Additive model tests
(a) compute the weight wk i = 1
g ′ (µ k−1
ωi
i ) V (µ k−1
i ) ,
(b) evaluate the pseudo data
with η k−1
i
= g(µk−1),
i
z k i = g ′ (µ k−1
i
s(speed,1.48)
s(speed,1.52)
-40 -20 0 20 40
-40 -20 0 20 40
)(yi − µ k−1
) + η k−1
,
(c) minimize with respect to β the least square objective
with W = diag(w1, . . . , wn). We get β k .
||W (z − Xβ)|| 2 +
λjβ T Sjβ,
(d) prepare next estimate with η k = Xβ k and µi = g −1 (η k i ).
1.8.2 R outputs
Bronchitis dataset
i
j
i
B-basis regression (GCV)
5 10 15 20 25
speed
B-basis regression (REML)
5 10 15 20 25
speed
1.8. Appendix
Let us study the example of Bronchitis data of Turner (2008). The data consists of 212
patients, on which we measure the presence/absence of bronchitis B for bron, the air pollution
level in the locality of residence P for poll and the number of cigarettes smoked per day C
for cigs, see Appendix 1.8.2.
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