tail order

[withoutapaddle]

the notes discuss the different loss distributions

can anyone tell me the order of each of the tails i.e o(x),o(n^-x) etc

namely for:

1. logNormal
2. pareto => o{x^-(alpha+1)}??
3. weibull
4. burr => o(x^-alpha)
5. exponential =>o{exp(-lambda*x)}??

and to make it blatently obvious for me, to put them in order of the tail which tends to zero fastest

i.e. o(x^-2) tends to zero faster than o(x^-1)

would looking at kurtosis give me a good estimate??

 

[Anna Bishop]

Hello w-a-p

I'm attaching a pdf of a document that I use in tutorials to demonstrate some of the tails. It contains all the distributions that you are after except for the Burr, and this is because the Burr is very tough to fit.

In the attachment are three graphs, each showing a different stage of the tail of the claim amount distributions.

To draw the graphs, I have fixed the mean at 500 and the sd at 774 for each distribution. I've then used method of moments to find the parameters of each distribution. I've then plotted the probability density function (f(x)) for each.

I expect the results are sensitive to the choice of mean and std deviation, but you should be able to see that, in the extreme, the lognormal seems to have the fattest tail and the exponential the thinest tail, with the other distributions in between.
View attachment Loss distributions.pdf

Anna