Tail Dependence - ETH - Entrepreneurial Risks - ETH Zürich

Setting all the xi’s but one to +∞ in equation (2.3) yields:
n
lim Fi (an,i + bn,i · xi) = Gi (xi) , fori = 1, . . ., d, (2.7)
n→∞
where Fi and Gi are the i-th marginals of F and G. In turn Fi ∈ MDA (Gi), where Gi
is a Gumbel, Fréchet, or Weibull distribution 1 .
In order to isolate the dependence features from the marginal distribution aspects [6]
(2000), traditionally the components of both the distribution F and the corresponding
MEV distribution G are transformed to standard marginals. It is customary to choose
the standard Fréchet distribution as marginals i.e. the function φ : R → [0, 1] given
by φ(x) = exp � �
−1 for x > 0 and zero elsewhere and by its inverse function: y =
x
φ−1 (x) = −1/ log(x), where ’log’ denotes the natural logarithm. Defining X as a
d-variate random row-vector with distribution F and continuous marginals
Y = φ −1 1
(F(X)) = − , (2.8)
log F(X)
i.e. Yi = φ−1 (Fi(Xi)) = −1/ log F(Xi) for all i ≤ 1 ≤ d. By the Probability Integral
Transform it follows that Y has standard Fréchet marginals. Letting G be a
multivariate distribution with continuous marginals Gi’s and defining:
G ∗ �� � (−1) � � �
(−1)
−1
−1
(y1, . . .,yd) = G
· yi, . . .,
· yd , (2.9)
log G1
log Gd
with y1 > 0, . . .,yd > 0, then G ∗ has standard Fréchet marginals, and G is a MEV distribution
only if G ∗ is also MEV distributed. Thus the marginals of a MEV distribution
can be standardized, yet preserving the extreme value properties.
2.2 The Survival Function
The survival function F associated with multivariate extreme value theory is given by:
F = Pr {X1 > x1, . . .,Xd > xd} (2.10)
In the univariate case d = 1 F = 1 − F(x). Unfortunately this does not hold generally
in multivariate EVT [5] (1988).
2.3 Multivariate Dependence
In the multivariate case the notions of dependence becomes numerous and complex.
Below we introduce some basic concepts. For further information about how to measure
dependence between random variables we refer to [7] (2006), [8] (2001), and [9] (1997).
2.3.1 Positive Orthant Dependence
Assuming that X = (X1, . . .,Xd) is a d-variate random row-vector, X is positively
lower orthant dependent (PLOD), if for all x = (x1, . . .,xd) ∈ R d ,
Pr {X ≤ x} ≥
d�
Pr {Xi ≤ xi} (2.11)
i=1
1 This is shown in Theorem 1.7 of chapter 1.2 of [3] (2007)
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