  Asymptotic normality of extreme value tail estimators received much attention
in the literature, giving rise to increasingly complicated 2nd order regularity
conditions. However, such conditions are really difficult to be checked for
real data. Especially it is difficult or impossible to check such conditions
using small samples. Beside that most of those conditions suffer from the
drawback of a potentially singular integral representations. However, we can
have various orders of approximation by normal distributions, e.g. Berry-Essen
Types and Edgeworth types. In this paper we indicate that for Berry-Essen Types
of normal approximation and related asymptotic normality of generalized Hill
estimators, we do not necessarily need 2nd order regularity conditions and we
can apply only Karamata's representation for regularly varying tails. 2nd order
regularity conditions however better relates to Edgeworth types of normal
approximations, albeit requiring larger data samples for their proper check.
Finally both expansions are prone for bootstrap and other subsampling
techniques. All existing results indicate that proper representation of tail
behavior play a special and somewhat intriguing role in that context. We dispel
that widespread opinion by providing a full characterization and
representation, in a general regular variation context, of the integral
singularity phenomenon, highlighting its relation to an asymptotical normality
of the Generalized Hill estimator without the 2nd order condition. Thus
application of this new methodology is simple and much more flexible, optimal
for real data sets. Alternative and powerful versions of the Hill plot are also
introduced and illustrated on ecological data of snow extremes from Slovakia.
