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I think you should be able to do "nice linear parametric [...] computations" with any member of the exponential family (of which the Gaussian is the best-known example).

Can the last politician to go out the revolving door please turn the lights off?
by Carrie (migeru at eurotrib dot com) on Fri Jun 22nd, 2007 at 09:28:28 AM EST
[ Parent ]
Well, I'm not familiar with the generalized exponential distribution framework, but from the primer referenced by the Wikipedia (bold mine):

For most insurance applications, we would expect a decreasing unit hazard functton. That is, as we move to higher and higher layers, the chance of a partial loss would decrease. For instance, if we consider a layer such as $10,000,000 xs $990,000,000 we would expect that any loss above $990,000,000 would almost certainly be a full-limit loss. This would imply h (y) ~ 0.

The decreasing hazard function is not what we generally find in the exponential family. For the Normal and Poisson, the hazard function approaches 1, implying that full-limit losses become less likely on higher layers - exactly the opposite of what our understanding of insurance phenomena would suggest. The Negative Binomial, Gamma and Inverse Gaussian distributions asymptotically approach constant amounts, mimicking the behavior of the exponential distribution

The table below shows the asymptotic behavior as we move to higher attachment points
for a layer of width w.

DistributionLimiting Form of h (y)Comments
Normallim h(y) = 1No loss exhausts the limit
Poissonlim h (),) = I
Negative Bmomiallim h,(y) = I - ( I - p ) "
Gammalim h(y) = I-e "''°'~'
Inverse Gausstanlim h(y) = I-e -''a°''~
Lognormallim h(v) = 0Every loss ts a full-limit loss

From this table, we see that the members of the natural exponential family have tail behavior that does not fully reflect the potential for extreme events in high casualty insurance. It would seem that the natural exponential distributions used with GLM are more appropriate for insurance lines without much potential for extreme events or natural catastrophes.

Sorry the layout is crappy, had to rework an OCR-ed pdf.
But the general idea seems to be that if it's linear, then you don't have a fat tail. I don't know if there is any kind of proof of this or of something heuristically similar (I'm by no mean a statistician)


by Pierre on Fri Jun 22nd, 2007 at 09:58:04 AM EST
[ Parent ]


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