Fourth class of extreme-value distributions? The generalized extreme-value distribution encompasses three classes of distributions:

*

*Frechet, which are regularly varying, infinite right limit.

*Gumbel, which are not regularly varying, infinite right limit.

*Weibull, which are regularly varying, finite right limit.

Is there a fourth class, one that fits in between the Gumbel and Weibull classes: which are not regularly varying, finite right limit?
If so, what is this class called?
And, is there a generalization of the generalized extreme-value distribution that encompasses this 4th class of distribution?
 A: No. The extreme-value theorem (Fisher/Tippett/Gnedenko) gives the possible limits of a distribution of maxima (appropriate scaled),  and they divide into three groups based on whether the extreme value index parameter is positive, zero, or negative. The generalised extreme value distribution is precisely the set of possible limits, and the three named subsets correspond to the positive, zero, and negative values of the index
A: Reading various lecture notes and corresponding with colleagues outside this forum, I now understand that, as Thomas Lumley reminded me, the extreme value theorem (Fisher/Tippett/Gnedenko) only divides into three cases (Frechet, Gumbel, Weibull). There is not an additional case tucked in between the Gumbel and Weibull that accommodates samples from source distributions that are right-limited, not regularly varying. Instead, these distributions are asymptotically like the Gumbel distribution. What continues to amaze me, however, is that while the samples might be right-limited, the "lightness" of the tail (apparently) permits description by a distribution (Gumbel) that is not right-limited. I will continue to ponder this point, but for now, I consider this question closed.
