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It would be highly inefficient to perform estimates merely on a few extreme values. If one estimates parameters using all threshold exceedances however, then there's a lot more data available to use. That's the idea anyway, together with the fact that the possible asymptotics belong to on of three families of EV distributions: Gumbel, Frechet or Negative Weibull.

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$E(X_t|F_s) = X_s,\quad t > s$
by martingale on Mon Sep 22nd, 2008 at 05:00:21 AM EST
[ Parent ]
The basic argument against the practicality of extreme value theory is this:
A Simple Proof Of Unpredictability In The Fourth Quadrant

I show elsewhere that if you don't know what a "typical" event is, fractal power laws are the most effective way to discuss the extremes mathematically. It does not mean that the real world generator is actually a power law--it means you don't understand the structure of the external events it delivers and need a tool of analysis so you do not become a turkey. Also, fractals simplify the mathematical discussions because all you need is play with one parameter (I call it "alpha") and it increases or decreases the role of the rare event in the total properties.

Um, forget about "fractal" - Taleb is in thrall to Mandelbrot. He's simply talking about power-law (and hence scale free and so "fractal") fat tails - e.g., Pareto. "Alpha" is the exponent of the power law
For instance, if you move alpha from 2.3 to 2 in the publishing business, the sales of books in excess of 1 million copies triple!  Before meeting Benoit Mandelbrot, I used to play with combinations of scenarios with series of probabilities and series of payoffs filling spreadsheets with clumsy simulations; learning to use fractals made such analyses immediate. Now all I do is change the alpha and see what's going on.
Fair enough, writing for laymen. Here's the rub:
Now the problem: Parametrizing a power law lends itself to monstrous estimation errors (I said that heavy tails have horrible inverse problems). Small changes in the "alpha" main parameter used by power laws leads to monstrously large effects in the tails. Monstrous.
This is a more jargon-laden restamement of if you move alpha from 2.3 to 2 in the publishing business, the sales of books in excess of 1 million copies triple!
And we don't observe the "alpha. Figure 5 shows more than 40 thousand computations of the tail exponent "alpha" from different samples of different economic variables (data for which it is impossible to refute fractal power laws). We clearly have problems figuring out what the "alpha" is: our results are marred with errors. Clearly the mean absolute error is in excess of 1 (i.e. between alpha=2 and alpha=3). Numerous papers in econophysics found an "average" alpha between 2 and 3--but if you process the >20 million pieces of data analyzed in the literature, you find that the variations between single variables are extremely significant.
The fact is that, assuming independent, identically distributed observations, the probability that the next observation will be larger than the past N observations is 1/(N+1) - with no upper limit for the confidence interval. That's the non-parametric view. The problem with the parametric view is that with power-law fat tails, as taleb correctly points out, small estimation errors on the parameter lead to huge variations in the confidence interval for the next observation conditional on it exceeding all previous observations.

A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
by Migeru (migeru at eurotrib dot com) on Tue Sep 23rd, 2008 at 05:54:05 AM EST
[ Parent ]
Wikipedia: Pareto distribution and Zipf's law.

A vivid image of what should exist acts as a surrogate for reality. Pursuit of the image then prevents pursuit of the reality -- John K. Galbraith
by Migeru (migeru at eurotrib dot com) on Tue Sep 23rd, 2008 at 06:01:12 AM EST
[ Parent ]

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