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

tel-00703797, version 2 - 7 Jun 2012
Binary regression and model selection
1.6. Other regression models
As for GLMs, the binary regression means we assume that Yi follows a Bernoulli distribution
B(πi), πi being linked to explanatory variables. So, the model equation is
πi = g −1 (ηi),
where g is the link function and ηi the predictor. Unlike the GLM where the predictor was
linear, for GAMs the predictor is a sum of smooth functions:
p
p1
p2
α0 + fj(Xj) or α0 + αiXi + fj(Xj),
j=1
the latter being a semi-parametric approach. As suggested in Hastie and Tibshirani (1995),
the purpose to use linear terms can be motivated to avoid too much smooth terms and are
longer to compute (than linear terms). For instance, if a covariate represents the date or the
time of events, it is “often” better to consider the effect as an increasing or decreasing trend
with a single parameter αi.
As for GLMs, we are able to compute confidence intervals using the Gaussian asymptotic
distribution of the estimators. The variable selection for GAMs is similar to those of GLMs.
The true improvement is a higher degree of flexibility to model the effect of one explanatory
variables on the response.
The procedure for variable selection is similar to the backward approach of GLMs, but a
term is dropped only if no smooth function and no linear function with this term is relevant.
That is to say, a poor significance of a variable modelled by a smooth function might be
significant when modelled by a single linear term. We will use the following acceptance rules
of Wood (2001) to drop an explanatory variable:
(a) Is the estimated degrees of freedom for the term close to 1?
(b) Does the plotted confidence interval band for the term include zero everywhere?
(c) Does the GCV score drop (or the REML score jump) when the term is dropped?
If the answer is “yes” to all questions (a, b, c), then we should drop the term. If only question
(a) answer is “yes”, then we should try a linear term. Otherwise there is no general rule to
apply. For all the computation of GAMs, we use the recommended R package mgcv written
by S. Wood.
1.6.2 Application to the large dataset
In Section 1.3.2, the GLM analysis of this large dataset reveals that the channel distribution
strongly impacts the GLM outputs. Especially, the lapse gap between tied-agent and other
channels is far stronger than what we could expect. Moreover, the price sensitivity gap
measured by the lapse deltas is also high. Let us see this it still holds with GAM results.
On each channel and cover, we first estimate a GAM by modelling all the terms by a
smooth function. And then we apply the Wood’s rules to remove, to linearize or to categorize
the explanatory variables. In Appendix 1.8.2, we provide the regression summary for one of
the nine subsets.
Comments on regression summary
In this subsection, we briefly comment on the nine regression summaries. Let us start with
the Third-Part Liability cover. For the agent subset, for which we have a market proxy, we
i=1
j=1
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