13-Fat Tails-Scaling-Stabl Page 367 Wednesday, February 4, 2004 1:00 PM
Fat Tails, Scaling, and Stable Laws 367
The MDA of the Weibull Distribution
The Weibull distribution is written as follows:
Ψα = exp[–(–x –α)]
The Weibull and the Frechet distributions are closely related to each
other. In fact, it is clear from the definition that the following relation-
ship holds:
Ψα()x = Φα(–x –^1 ), x > 0
One can therefore expect that the MDA of the two distributions are
closely related. In fact, it can be demonstrated that a distribution func-
tion F belongs to the MDA of a Weibull distribution α> 0 if and only if
xF < ∞
and
Fx( F – x –^1 )= x –αLx()
where L is a slowly varying function.
If
F ∈MDA(Ψα)
then
c–^1 (M – x D
n n F) → Ψα
The MDA of the Weibull distribution includes important distribu-
tions such as the distribution uniform in (0,1), power laws truncated to
the right, and Beta distributions.
The MDA of the Gumbel Distribution
The Gumbel distribution is written as Λ(x) = exp[–exp(–x)]. Observe
that the Gumbel distribution has exponential tails. This fact can be eas-
ily ascertained through Taylor expansion. There is no simple character-
ization of the MDA of the Gumbel Distribution.