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?