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.

Distribution	Limiting Form of h (y)	Comments
Normal	lim h(y) = 1	No loss exhausts the limit
Poisson	lim h (),) = I
Negative Bmomial	lim h,(y) = I - ( I - p ) "
Gamma	lim h(y) = I-e "''°'~'
Inverse Gausstan	lim h(y) = I-e -''a°''~
Lognormal	lim h(v) = 0	Every 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)

Pierre