The Mathematics of Financial Modelingand Investment Management

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.