13-Fat Tails-Scaling-Stabl Page 368 Wednesday, February 4, 2004 1:00 PM
368 The Mathematics of Financial Modeling and Investment Management
The MDA of a Gumbel distribution encompasses a large class of dis-
tributions that includes the exponential distribution, the normal distribu-
tion, and the lognormal distribution. Though the Gumbel distribution
has exponential tails, its MDA includes subexponential distributions
such as the Berktander distribution, as explained in Goldie and Resnick.^7
Max-Stable Distributions
Stable distributions remain unchanged after summation; max-stable dis-
tributions remain unchanged after taking maxima. A non-degenerate
random variable X and the relative distribution is called max-stable if
there are constants cn > 0, dn ∈ R such that the following conditions are
satisfied
D
max(X 1 , ..., Xn) = c Xdn + n
where X, X 1 , ..., Xn are IID variables.
It can be demonstrated that the class of max-stable distributions
coincides with the class of possible limit laws for normalized and cen-
tered maxima. In view of the previous discussions, the max-stable laws
are the three possible limit laws: Frechet, Weibull, and Gumbel.
Generalized Extreme Value Distributions
The three extreme value distributions, Frechet, Weibull, and Gumbel,
can be represented as a one-parameter family of distributions through
the Standard Generalized Extreme Value Distribution (GEV) of Jenkin-
son and Von Mises. Define the distribution function Hξ as follows:
exp[–( 1 + ξx) –^1 ⁄ ξ] for ξ ≠ 0
Hξ =
exp(–exp(–x)) for ξ =^0
where 1 + ξx > 0. One can see from the definition that ξ = α–1 > 0 corre-
sponds to the Frechet distribution, ξ = 0 corresponds to the Gumbel dis-
tribution, and ξ = –α–1 < 0 corresponds to the Weibull distribution. We
can now introduce the related location-scale dependent family Hξ;μ,ψ by
replacing the argument x with (x – μ)/ψ.
(^7) C.M. Goldie and S. Resnick, “Distributions that are Both Subexponential and in
the Domain of Attraction of an Extreme-Value Distribution,” Advanced Applied
Probability, 20 (1988), pp. 706–718.