Modelling Dependence with Copulas - IFOR

7 Extreme Value Copulas
7.1 Univariate Extreme Value Theory
Let X 1 ,X 2 ,... be iid random variables with continuous distribution function F .
Let S n = X 1 + ... + X n ,M n =max(X 1 ,... ,X n )andL n =min(X 1 ,... ,X n ).
Furthermore let x F ,givenby
x F =sup{x ∈ R : F (x) < 1} ≤∞
x
denote the right endpoint of F . It can be shown that M n → x F as n →∞. If X i
has finite mean and variance µ and σ 2 respectively, we know from the central limit
theorem that
P[(S n − nµ)/ √ nσ 2 ≤ x] → Φ(x) asn →∞,
where Φ is the distribution function of the standard normal distribution. What
about normalized maxima and minima? Consider possible limiting distributions
for (M n − a n )/b n and (L n − c n )/d n as n → ∞ for suitably chosen sequences
{a n }, {b n }, {c n }, {d n }. Since
min(X 1 ,... ,X n )=− max(−X 1 ,... ,−X n )
it is sufficient to study the case of maxima. The sequences {a n } and {b n } we are
interested in are such that
P[(M n − a n )/b n ≤ x] =P[M n ≤ a n + b n x]=F n (a n + b n x) → H(x),
as n →∞for some non-degenerate distribution function H, i.e. {H(x)|x ∈ R} ̸=
{0, 1}. The three possible types for H are called univariate (max) extreme value
distributions and are given by:
• Fréchet
{
0, x ≤ 0
φ α (x) =
exp(−x −α ), x > 0 α>0.
• Gumbel
Λ(x) =exp(−e −x ),
x ∈ R
• Weibull
Ψ α (x) =
{
exp(−(−x) α ), x ≤ 0
1, x > 0 α>0.
We say that two random variables U and V are of the same type if and only if
This means that
U = d aV + b for b>0,a∈ R.
F U (x) =F V ((x − b)/a)
and hence random variables of the same type have the same distribution function
up to changes of scale and location. Thus in the non-degenerate case the only
possible limits of (M n − a n )/b n are the location-scale families based on the above
three distribution functions. In this case we say that F is in the maximum domain
of attraction of H,
F ∈ MDA(H).
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