Hi Peggy,

On the first:

Another nice observation! If I had to point to one weakness in the 2008 AIMs (vs. 2007), it would be the tendency to omit some foundational ideas. Last year, there was a much better setup to for this in the Quant.

If you roll a six-sided die twice and get, say, a '2' and a '5', and call it a day, this histogram can be an *empirical* PMF distribution: P[1=0,2=50%,3=0,4=0,5=50%, and 6=0%]. Empirical = based on the raw outcomes. As opposed to the corresponding parametric PMF: P[X]=1/6.

Gujarati 2.4 contrasts classical (a priori) with empirical (relative) probability. The first "maps" to a parametric distribution; the latter to an empirical distribution.

"...while the tails where there is a shortage of data, parametric distributions like the EVT Pots over threshold"

So this is a key problem is risk measurement, and maybe the key problem in

*operational* risk measurement: we'd love an accurate empirical distribution (P[loss @ 99%] = based on historical actual distribution) for the extreme loss tail, but that is exactly where we lack data, so we

**probably must settle for parametric**. (It is stunning how much research has been devoted to this: finding a parametric distribution where there is no data to springboard from). We are comfortable using empirical data/distribution near the center, but we aren't focused there. Much like the two rolls example above, you don't trust my two-rolls-empirical distribution b/c it's based on so few rolls. If i rolled 100 times, maybe you'd trust it. So, in the extreme tail, where events by definition are "low frequency, high severity" we are pretty much forced to go with parametric distributions. In some respect, I'd be inclined to characterize the whole EVT approach as: "we don't have piles and piles of good data in the tail, so let us graft onto the tail these sets of parametric distributions."

Finally, another idea at work here is the idea the full (CDF) distribution can be generated by stitching together more than one sub-distributions. In this case, an empirical in the middle gives way to a parametric for the tail (you can see, as i've said, we may expect the tail to be "forced" into a parameteric). Two days ago I blogged about an interesting, helpful paper that takes this typical MIXTURE MODEL approach:

http://www.bionicturtle.com/learn/article/operational_loss_dependencies_academic_paper/
Here is an example of a mixture model, I will graft a coin toss distribution onto a six-sided die distribution: P[X = 1 to 6 = 50% * 1/6; X = 7 or 8 = 50% * 1/2]. And now i have made my own sort of fat tail. You are referring to a mixture-model where the start is empirical and then "graft" or "stich" an EVT parametric.

The EVT reading this year is far too scant. (Dowd's book Model Risk has nice chapter). The two EVT distributions that Wilmott briefly touches on (GPD, GEV) are, by definition, part of a mixture model: they characterize the tail so they are attached/grafted onto something else for the middle/body (e.g., CDF P[X] < 60% or 70% or 80%). These EVT distributions attach to the tail: they start at, say, 99%. Again, for CDF P() < 95% or < 99%, could use a normal distribution. Then start an EVT for CDF P(99% < X < 100%). See how the whole EVT distribution is both parameteric (no data, need formula) and "mixed" into the tail at the end?

David

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