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I'm trying to understand the advantages (if any) of employing the Generalized Extreme Value distribution (GEV) vs. a stable distribution in the context of understanding the probability of crossing a large threshold.

The stable is limiting for averages, whereas the GEV is limiting for extremes. Both distributions have heavy-tailed forms (relevant for crossing high threshold).

But it seems to me that the one should employ the GEV because 1) it focuses on the largest values, which are of interest; 2) it can describe data w/ more types of tailedness (can have a bounded upper tail, while the stable can only be heavy or thin). Are these differences correct?

Assume large values and thresholds of are interest, and someone has analyzed a time series and found that it follows a Lévy-stable distribution (i.e., is heavy-tailed). Is it sufficient (or satisfactory) to to draw conclusions about the probability of observing a certain extreme value using the stable, or is there more information to be gained by using the GEV?

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one interesting about stables: they're stable :)

the sum of stable is a stable distribution. it's a very handy property in many applications. for instance, you can create random walk like processes with them, and they scale over time. when you split time into smaller intervals, they still have the same distribution. it's very convenient when estimating error variance of forecast

they have problems too. some of them have no moments, e.g. Cauchy distribution. it doesn't stop physicists from using them, but some mathematicians freak out.

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  • $\begingroup$ does the stable distribution still hold if you apply the "maximum" function to portions of the time series? $\endgroup$ – rbatt Mar 4 '14 at 23:50

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